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Module homomorphism

In abstract algebra, a module homomorphism (or R-module homomorphism) is a structure-preserving map between two modules over the same ring R. Specifically, given left R-modules M and N, a function φ: M → N is an R-module homomorphism if it is a group homomorphism of the underlying abelian groups and satisfies φ(r m) = r φ(m) for all r ∈ R and m ∈ M.^[1] This condition ensures that φ respects both addition in the modules and scalar multiplication by ring elements, generalizing the notion of linear maps between vector spaces when R is a field.^[2] Key properties of module homomorphisms include their kernels and images, which are themselves submodules. The kernel of φ, denoted ker φ, is the submodule {m ∈ M | φ(m) = 0}, consisting of elements mapped to the zero element in N.^[3] The image of φ, denoted im φ, is the submodule {φ(m) | m ∈ M} ⊆ N, representing the submodule generated by the outputs of φ.^[3] These submodules play a central role in the first isomorphism theorem for modules, which states that if K = ker φ, then the quotient module M/K is isomorphic as an R-module to im φ via the natural map induced by φ.^[2] Module homomorphisms are fundamental in homological algebra, where they form the morphisms in the category of R-modules.^[4] They enable the construction of exact sequences, such as the short exact sequence 0 → ker φ → M → im φ → 0 associated to any homomorphism φ: M → N, which captures relationships between modules via injectivity and surjectivity conditions.^[4] This framework underpins concepts like projective and injective modules and Ext functors, with applications such as in algebraic geometry.^[5]

Fundamentals

Definition

In module theory, a branch of abstract algebra, a module homomorphism (also known as an R-linear map) is a structure-preserving map between two modules over the same ring. Specifically, let R be a ring with identity, and let M and N be R-modules (typically left R-modules unless otherwise specified). An R-module homomorphism φ: M → N is a function φ that satisfies two conditions: it is additive, meaning φ(m₁ + m₂) = φ(m₁) + φ(m₂) for all m₁, m₂ ∈ M, and it respects scalar multiplication, meaning φ(r m) = r φ(m) for all r ∈ R and m ∈ M.^[2]^[1] Equivalently, since every R-module is an abelian group under addition, φ is a group homomorphism of the underlying abelian groups that commutes with the R-action. This definition generalizes the notion of a linear transformation between vector spaces, where R is a field; in that case, every module homomorphism is a linear map.^[3]^[6] The set of all R-module homomorphisms from M to N, denoted Hom_R(M, N), forms an abelian group under pointwise addition: (φ + ψ)(m) = φ(m) + ψ(m) for φ, ψ ∈ Hom_R(M, N) and m ∈ M. If R is commutative, Hom_R(M, N) is itself an R-module via (r φ)(m) = r φ(m). A homomorphism φ is called an isomorphism if it is bijective (hence invertible, with inverse also a homomorphism), a monomorphism if it is injective, and an epimorphism if it is surjective.^[1]^[7]

Terminology

A module homomorphism between two modules over a ring R is a function f: M \to N that preserves both the abelian group structure and the scalar multiplication by elements of R, satisfying f(x + y) = f(x) + f(y) for all x, y \in M and f(r x) = r f(x) for all r \in R and x \in M.^[8]^[3]^[9] The set of all R-module homomorphisms from M to N, denoted \mathrm{Hom}_R(M, N) or simply \mathrm{Hom}(M, N) when the ring is clear from context, forms an abelian group under pointwise addition, defined by (f + g)(x) = f(x) + g(x) for f, g \in \mathrm{Hom}_R(M, N) and x \in M.^[9] If R is commutative, \mathrm{Hom}_R(M, N) inherits an additional R-module structure via (r f)(x) = r f(x).^[9] Special cases of module homomorphisms include monomorphisms, which are injective maps; epimorphisms, which are surjective maps; and isomorphisms, which are bijective homomorphisms with bijective inverses that are also homomorphisms.^[3] An endomorphism is a homomorphism from a module to itself, so f: M \to M, and the set of endomorphisms \mathrm{End}_R(M) forms a ring under composition.^[9] An automorphism is an isomorphism from a module to itself, and the set of automorphisms \mathrm{Aut}_R(M) forms a group under composition.^[3] The zero homomorphism, or trivial homomorphism, is the map sending every element of M to the zero element of N, which is always a valid homomorphism.^[8] When R is a field, module homomorphisms coincide with linear transformations between vector spaces.^[8]^[9]

Basic Properties

Kernel and Cokernel

In the category of modules over a ring R, the kernel of an R-module homomorphism f: M \to N is defined as the submodule \ker(f) = \{ m \in M \mid f(m) = 0 \}.^[10] This set is an R-submodule of M, as it is the kernel of f when viewed as an abelian group homomorphism, and the submodule property follows from the R-linearity of f.^[10] Similarly, the image \operatorname{im}(f) = \{ f(m) \mid m \in M \} is an R-submodule of N.^[3] The first isomorphism theorem for modules states that M / \ker(f) \cong \operatorname{im}(f) as R-modules, where the isomorphism is induced by the canonical projection M \to M / \ker(f) composed with the map sending the coset m + \ker(f) to f(m).^[10] This quotient construction shows that \ker(f) measures the "failure" of f to be injective, and the theorem provides a canonical way to identify the image with a quotient module. The cokernel of f: M \to N is defined as \operatorname{coker}(f) = N / \operatorname{im}(f), which is an R-module via the quotient structure.^[3] The natural projection \pi: N \to \operatorname{coker}(f) given by n \mapsto n + \operatorname{im}(f) is a surjective R-module homomorphism with kernel \operatorname{im}(f).^[10] Thus, \operatorname{coker}(f) captures the "failure" of f to be surjective, and the exact sequence

$0 \to \operatorname{im}(f) \to N \to \operatorname{coker}(f) \to 0

holds, where the first map is the inclusion and the second is \pi.^[3] In homological algebra, f fits into the exact sequence

$0 \to \ker(f) \to M \xrightarrow{f} N \to \operatorname{coker}(f) \to 0,

which is exact at M and N by the definitions of kernel and cokernel; exactness at \ker(f) requires the inclusion to be the kernel map.^[3] This sequence universalizes the properties: any homomorphism factoring through f extends uniquely over the cokernel, and similarly for the kernel. For example, if R = \mathbb{Z}, then for f: \mathbb{Z} \to \mathbb{Z} given by multiplication by 2, \ker(f) = \{0\} and \operatorname{coker}(f) = \mathbb{Z}/2\mathbb{Z}.^[10]

Image and Exactness

The image of an R-module homomorphism f: M \to N is the submodule \operatorname{Im}(f) = \{f(m) \mid m \in M\} \subseteq N.^[10] This submodule consists of all elements in N that are reachable from M under f, and it inherits the module structure from N.^[10] A key property is that f factors through the quotient module M / \ker(f), establishing a canonical isomorphism \operatorname{Im}(f) \cong M / \ker(f), known as the first isomorphism theorem for modules.^[2] This theorem implies that the image captures the "essential" action of f, modulo elements mapped to zero, and holds for any ring R and modules M, N.^[2] Exactness provides a framework for relating images and kernels in sequences of module homomorphisms. A sequence of R-modules and homomorphisms \dots \to M_{i-1} \xrightarrow{f_{i-1}} M_i \xrightarrow{f_i} M_{i+1} \to \dots is exact at M_i if \operatorname{Im}(f_{i-1}) = \ker(f_i).^[11] This condition ensures that every element in the kernel of the outgoing map arises precisely from the image of the incoming map, with no overlaps or deficiencies.^[11] For instance, in the short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0, exactness at A and C implies f is injective and g is surjective, while exactness at B gives \operatorname{Im}(f) = \ker(g), yielding C \cong B / A.^[3] Such sequences are fundamental in homological algebra, as they encode extensions and relations between modules. For example, the first isomorphism theorem can be rephrased as the exactness of $0 \to \ker(f) \to M \to \operatorname{Im}(f) \to 0.^[12] Properties like the five lemma further leverage exactness to preserve isomorphisms in commutative diagrams of short exact sequences.^[3]

Examples

Elementary Examples

One of the simplest module homomorphisms is the zero map, which sends every element of a module M to the zero element in another module N, preserving addition and scalar multiplication by mapping all sums and scalar multiples to zero.^[3] Similarly, the identity map from a module M to itself acts as f(m) = m for all m \in M, which is both a homomorphism and an isomorphism since it is bijective and respects module operations.^[3] For \mathbb{Z}-modules, which coincide with abelian groups under the module structure where scalar multiplication by n \in \mathbb{Z} is repeated addition, any group homomorphism is automatically a module homomorphism.^[1] A concrete example is the map \phi: \mathbb{Z} \to \mathbb{Z} defined by \phi(k) = 2k, which preserves addition (\phi(k + l) = 2(k + l) = 2k + 2l = \phi(k) + \phi(l)) and scalar multiplication (\phi(n \cdot k) = \phi(nk) = 2(nk) = n(2k) = n \cdot \phi(k)).^[13] Another example between cyclic modules is a homomorphism \psi: \mathbb{Z}/6\mathbb{Z} \to \mathbb{Z}/3\mathbb{Z} sending the generator $1 \mod 6 to $0 \mod 3, which extends to \psi(k \mod 6) = 0 for all k, as it must annihilate elements of order dividing 6 while respecting the relations in the codomain.^[14] When the ring is a field k, modules are vector spaces, and homomorphisms are precisely linear transformations.^[15] For instance, the projection map from k^2 to k given by (x, y) \mapsto x is a k-module homomorphism, as it preserves vector addition and scalar multiplication by elements of k: \phi((x_1, y_1) + (x_2, y_2)) = x_1 + x_2 = \phi(x_1, y_1) + \phi(x_2, y_2) and \phi(c(x, y)) = cx = c \cdot \phi(x, y).^[15] Over the polynomial ring k where k is a field, an elementary example is the shift map f: k \to k defined by f(p(x)) = x p(x), which is a k-module homomorphism because it preserves addition (f(p + q) = x(p + q) = xp + xq = f(p) + f(q)) and scalar multiplication by polynomials (f(r(x) p(x)) = x (r(x) p(x)) = r(x) (x p(x)) = r(x) f(p)).^[3] This map is injective but not surjective, illustrating non-isomorphic modules with homomorphisms between them.^[15]

Homomorphisms Between Free Modules

Free modules over a ring R are the module-theoretic analogs of vector spaces, consisting of direct sums of copies of R itself. A homomorphism \phi: R^m \to R^n between free modules of finite rank is uniquely determined by the images of the standard basis elements e_1, \dots, e_m of R^m, which can be expressed as \phi(e_j) = \sum_{i=1}^n a_{ij} e_i for coefficients a_{ij} \in R. This representation corresponds to left multiplication by an n \times m matrix A = (a_{ij}) with entries in R, so that \phi(x) = A x for any column vector x \in R^m.^[16]^[17] The matrix representation depends on the choice of bases for the domain and codomain. If \{f_1, \dots, f_m\} and \{g_1, \dots, g_n\} are bases for R^m and R^n, respectively, then \phi is represented by the matrix [ \phi ]_{g,f} whose i-th column is the coordinate vector of \phi(f_i) with respect to the basis \{g_k\}. Changing bases via invertible matrices P \in \mathrm{GL}_m(R) and Q \in \mathrm{GL}_n(R), where the columns of P and Q are the coordinates of the new bases in the old bases for the domain and codomain respectively, transforms the matrix to Q^{-1} A P, preserving the isomorphism class of the homomorphism. This equivalence relation allows for the study of homomorphisms up to change of basis, analogous to similarity in linear algebra.^[16]^[18]^[19] Composition of homomorphisms between free modules corresponds to matrix multiplication. If \psi: R^k \to R^m and \phi: R^m \to R^n are represented by matrices B ( m \times k ) and A ( n \times m ), respectively, then \phi \circ \psi is represented by A B, an n \times k matrix. This holds provided R is commutative, ensuring that scalar multiplication is well-defined; in the non-commutative case, one must specify left or right modules and adjust for the ring action. Over principal ideal domains (PIDs) like \mathbb{Z} or k ( k a field), any such matrix can be brought to Smith normal form via elementary row and column operations, yielding a diagonal matrix \mathrm{diag}(d_1, \dots, d_r, 0, \dots, 0) where d_i divides d_{i+1} and each d_i is unique up to units. This form classifies finitely generated modules as direct sums involving cyclic components.^[16]^[17] For example, consider R = \mathbb{Z} and \phi: \mathbb{Z}^2 \to \mathbb{Z} given by \phi(a, b) = 2a + 3b, represented by the row matrix [2 \, 3]. The image is $2\mathbb{Z} + 3\mathbb{Z} = \mathbb{Z}, so \phi is surjective, and its kernel is generated by (-3, 2), yielding cokernel 0. In the vector space case over a field k, homomorphisms \phi: k^m \to k^n are precisely linear transformations, with the matrix A having rank equal to \dim(\mathrm{im} \phi). These examples illustrate how matrix invariants like determinant (over commutative rings with identity) detect isomorphisms: \phi is an isomorphism if and only if A is invertible, i.e., \det(A) is a unit in R.^[16]^[17]

Module Structures on Hom Spaces

Abelian Group Structure

The set \Hom_R(M, N) of all R-module homomorphisms from an R-module M to an R-module N is equipped with an abelian group structure under pointwise addition.^[20] For any f, g \in \Hom_R(M, N), the sum f + g is the homomorphism defined by (f + g)(m) = f(m) + g(m) for all m \in M.^[21] This addition is well-defined, as the sum of two R-linear maps is again R-linear: for r \in R and m \in M, (f + g)(r m) = f(r m) + g(r m) = r f(m) + r g(m) = r (f + g)(m), and it preserves the underlying abelian group structure of the modules.^[22] The identity element of this group is the zero homomorphism $0: M \to N, which sends every element of M to the zero element of N.^[23] For each f \in \Hom_R(M, N), the additive inverse -f is given by (-f)(m) = -f(m) for all m \in M, which is also an R-module homomorphism since scalar multiplication by -1 commutes with the R-action.^[20] Associativity follows from the associativity of addition in N: (f + g) + h = f + (g + h) pointwise, and commutativity holds because N is an abelian group, so f + g = g + f.^[4] This group structure is independent of the ring R and arises solely from the additive groups underlying M and N.^[21] For example, if M = R and N = R as left R-modules, then \Hom_R(R, R) is isomorphic to R as an abelian group via evaluation at $1 \in R, reflecting the ring's additive structure.^[24] In general, the abelian group \Hom_R(M, N) captures the linear maps between the modules while inheriting the additivity from their underlying abelian groups.^[22]

Module Structure over Endomorphism Rings

In the context of left R-modules M and N, the abelian group \mathrm{Hom}_R(M, N) of R-module homomorphisms acquires a natural left module structure over the endomorphism ring \mathrm{End}_R(M) = \mathrm{Hom}_R(M, M). For \alpha \in \mathrm{End}_R(M) and f \in \mathrm{Hom}_R(M, N), the action is defined by

(\alpha \cdot f)(m) = f(\alpha(m))

for all m \in M. This defines a ring action because composition in \mathrm{End}_R(M) is associative, and the action distributes over addition in both \mathrm{End}_R(M) and \mathrm{Hom}_R(M, N), preserving the R-linearity of homomorphisms.^[10] When R is commutative, \mathrm{End}_R(M) is an R-algebra, and the resulting left \mathrm{End}_R(M)-module structure on \mathrm{Hom}_R(M, N) is compatible with the canonical R-module structure on \mathrm{Hom}_R(M, N).^[10] Dually, \mathrm{Hom}_R(M, N) carries a right module structure over the endomorphism ring \mathrm{End}_R(N). For f \in \mathrm{Hom}_R(M, N) and \beta \in \mathrm{End}_R(N), the action is given by

(f \cdot \beta)(m) = \beta(f(m))

for all m \in M. This construction ensures associativity with respect to composition in \mathrm{End}_R(N) and compatibility with the additive group operation, making \mathrm{Hom}_R(M, N) a right \mathrm{End}_R(N)-module.^[25] The right action is R-linear when R is commutative, aligning with the left R-action on \mathrm{Hom}_R(M, N).^[25] Together, these endow \mathrm{Hom}_R(M, N) with the structure of an (\mathrm{End}_R(M), \mathrm{End}_R(N))-bimodule, where the left and right actions commute: (\alpha \cdot f) \cdot \beta = \alpha \cdot (f \cdot \beta) for \alpha \in \mathrm{End}_R(M), f \in \mathrm{Hom}_R(M, N), and \beta \in \mathrm{End}_R(N). This bimodule structure facilitates the study of module categories, as it allows endomorphisms to act naturally on morphism spaces, preserving exactness in certain functorial contexts such as adjoint isomorphisms.^[25] For instance, if M is projective, the left \mathrm{End}_R(M)-module \mathrm{Hom}_R(M, N) reflects properties of N through the action.^[10]

Representations

Matrix Representation

When the domain and codomain of a module homomorphism are free modules of finite rank over a ring R, the homomorphism admits a natural matrix representation relative to chosen bases. Specifically, let M \cong R^m and N \cong R^n, with bases \{e_1, \dots, e_m\} for M and \{f_1, \dots, f_n\} for N. A homomorphism \phi: M \to N is uniquely determined by the images \phi(e_j) for j = 1, \dots, m, each of which can be expressed as \phi(e_j) = \sum_{i=1}^n a_{ij} f_i with coefficients a_{ij} \in R. These coefficients form the columns of an n \times m matrix A = (a_{ij}) over R, such that \phi(x) = A x for any x \in M identified with column vectors in R^m.^[17]^[26] This matrix representation aligns with the R-module structure on the Hom space \mathrm{Hom}_R(M, N), which is isomorphic to the module of n \times m matrices over R. Addition of homomorphisms corresponds to matrix addition, and scalar multiplication by r \in R corresponds to multiplying the matrix by r. Composition of homomorphisms \phi: M \to N and \psi: N \to P \cong R^p is represented by matrix multiplication B A, where B is the matrix for \psi. Such representations are basis-dependent but change by invertible matrices under basis changes: if new bases are related by invertible matrices P and Q, the new matrix is Q^{-1} A P.^[17]^[26] For example, consider R = \mathbb{Z} and \phi: \mathbb{Z}^2 \to \mathbb{Z}^3 defined by \phi(1,0) = (2,0,0) and \phi(0,1) = (0,3,1). Relative to standard bases, the matrix is

A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \\ 0 & 1 \end{pmatrix}.

Then \phi(a,b) = (2a, 3b, b). This extends the familiar vector space case, where R is a field, but over general rings, the matrices need not be invertible even if \phi is an isomorphism.^[17] In the context of module presentations, homomorphisms from free modules can also define quotients via relation matrices. A surjective \phi: R^n \to M with finitely generated kernel presents M \cong R^n / \mathrm{im}(B), where B is the matrix whose columns generate \ker \phi. Over principal ideal domains, such matrices can be diagonalized via the Smith normal form to classify the module. However, for non-free modules, representations require choices of bases or generating sets, and may not be unique without additional structure.^[17]

Universal Properties

In the category of left R-modules, where R is a ring, module homomorphisms play a central role in defining universal properties of various constructions. A key example is the universal property of free modules. Let P be a free R-module with basis X. For any R-module M and any function f: X \to M, there exists a unique R-module homomorphism \tilde{f}: P \to M such that \tilde{f}|_X = f. This property characterizes free modules up to isomorphism and ensures that homomorphisms from free modules are determined solely by their action on basis elements.^[27] This universal property extends naturally to direct sums, which serve as coproducts in the category of R-modules. Consider a family of R-modules \{M_i\}_{i \in I}. The direct sum \bigoplus_{i \in I} M_i comes equipped with inclusion maps \iota_i: M_i \to \bigoplus_{i \in I} M_i. For any R-module N and any family of homomorphisms \{g_i: M_i \to N\}_{i \in I}, there exists a unique homomorphism g: \bigoplus_{i \in I} M_i \to N such that g \circ \iota_i = g_i for all i \in I. This property highlights how module homomorphisms facilitate the gluing of maps from summands into a single map from the coproduct.^[28] Dually, products provide limits in the category, with module homomorphisms defining the universal property. For the family \{M_i\}_{i \in I}, the direct product \prod_{i \in I} M_i has projection maps \pi_i: \prod_{i \in I} M_i \to M_i. Given any R-module N and homomorphisms \{h_i: N \to M_i\}_{i \in I}, there is a unique homomorphism h: N \to \prod_{i \in I} M_i satisfying \pi_i \circ h = h_i for all i. This ensures that homomorphisms into products are equivalently families of componentwise maps, underscoring the categorical balance between sums and products via homomorphisms.^[29] These properties extend to other constructions, such as quotients. The quotient module M/K, for a submodule K \subseteq M, satisfies: for any R-module N and homomorphism f: M \to N with f(K) = 0, there exists a unique homomorphism \overline{f}: M/K \to N such that \overline{f} \circ q = f, where q: M \to M/K is the canonical projection. Such universal characterizations rely fundamentally on the preservation of module structure by homomorphisms.^[30]

Operations on Homomorphisms

Addition and Scalar Multiplication

The set of all R-module homomorphisms from an R-module M to an R-module N, denoted \Hom_R(M, N), forms an abelian group under pointwise addition. For any \phi, \psi \in \Hom_R(M, N) and m \in M, the sum \phi + \psi is defined by

(\phi + \psi)(m) = \phi(m) + \psi(m).

This operation is well-defined because N is an abelian group under addition, and \phi + \psi preserves both addition in M and scalar multiplication by elements of R, since

(\phi + \psi)(m_1 + m_2) = \phi(m_1) + \psi(m_1) + \phi(m_2) + \psi(m_2) = (\phi(m_1) + \phi(m_2)) + (\psi(m_1) + \psi(m_2)) = \phi(m_1 + m_2) + \psi(m_1 + m_2)

and

(\phi + \psi)(r m) = \phi(r m) + \psi(r m) = r \phi(m) + r \psi(m) = r (\phi(m) + \psi(m))

for all r \in R and m_1, m_2 \in M. Thus, \phi + \psi \in \Hom_R(M, N). The identity element is the zero homomorphism sending every element of M to the zero element of N, and inverses are given by -\phi, where (-\phi)(m) = -\phi(m).^[31]^[22] In general, for non-commutative R, \Hom_R(M, N) is only an abelian group under addition and does not carry a natural R-module structure. However, if R is commutative, \Hom_R(M, N) inherits a scalar multiplication from R, making it a left R-module. For r \in R and \phi \in \Hom_R(M, N), define

(r \phi)(m) = r \cdot \phi(m)

for all m \in M. This is an R-module homomorphism because

(r \phi)(m_1 + m_2) = r \cdot \phi(m_1 + m_2) = r (\phi(m_1) + \phi(m_2)) = r \phi(m_1) + r \phi(m_2) = (r \phi)(m_1) + (r \phi)(m_2)

and, assuming R commutative,

(r \phi)(s m) = r \cdot \phi(s m) = r (s \phi(m)) = (r s) \phi(m) = (s r) \phi(m) = s (r \phi(m)) = s ((r \phi)(m))

for all s \in R and m_1, m_2 \in M. The operations satisfy the module axioms, such as distributivity: r (\phi + \psi) = r \phi + r \psi and (r s) \phi = r (s \phi), verified pointwise using the properties of N.^[31]^[15] When M = N, the set \End_R(M) = \Hom_R(M, M) of endomorphisms becomes a ring, with addition as above and multiplication given by composition of maps, but the additive group structure remains the same. For example, consider R = \mathbb{Z} (so modules are abelian groups) and M = N = \mathbb{Z}^2; then homomorphisms correspond to integer matrices, and addition of homomorphisms is matrix addition, while scalar multiplication by n \in \mathbb{Z} is multiplication of each matrix entry by n. This illustrates how the module structure on \Hom_R(M, N) generalizes the familiar vector space of linear maps over fields.^[31]^[22]

Composition

The composition of module homomorphisms is defined as follows: given R-module homomorphisms f: M \to N and g: N \to P, their composition g \circ f: M \to P is the function (g \circ f)(m) = g(f(m)) for all m \in M.^[3] To verify that g \circ f is itself an R-module homomorphism, consider the additivity property: for m_1, m_2 \in M,

(g \circ f)(m_1 + m_2) = g(f(m_1 + m_2)) = g(f(m_1) + f(m_2)) = g(f(m_1)) + g(f(m_2)) = (g \circ f)(m_1) + (g \circ f)(m_2),

using the additivity of f and g. Similarly, for scalar multiplication with r \in R,

(g \circ f)(r m) = g(f(r m)) = g(r f(m)) = r g(f(m)) = r (g \circ f)(m),

employing the R-linearity of both f and g. Thus, composition preserves the module structure.^[3]^[10] This operation endows the category of R-modules with a natural composition, forming a category \mathbf{Mod}_R where objects are R-modules and morphisms are homomorphisms. In particular, when M = P, the set \operatorname{End}_R(M) = \operatorname{Hom}_R(M, M) becomes a ring under pointwise addition and composition as multiplication, known as the endomorphism ring of M. The identity map \operatorname{id}_M: M \to M serves as the multiplicative identity in this ring.^[10] Composition interacts with other operations on homomorphisms; for instance, it is bilinear over the abelian group structure on \operatorname{Hom}_R(M, N), meaning (g_1 + g_2) \circ f = g_1 \circ f + g_2 \circ f and g \circ (f_1 + f_2) = g \circ f_1 + g \circ f_2, as well as distributive with respect to scalar multiplication. These properties facilitate the study of module morphisms in homological algebra.^[32]

Exact Sequences

Short Exact Sequences

In the context of module homomorphisms, a short exact sequence is a sequence of R-modules and R-module homomorphisms of the form $0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0, where iis injective,pis surjective, and\operatorname{im} i = \ker p.[34][33] This structure captures extensions of the module CbyA, meaning Bis constructed such thatAembeds as a submodule andCis isomorphic to the quotientB/A. Such sequences are central to homological algebra, as they encode relationships between modules via their homomorphisms and facilitate the study of derived functors like \operatorname{Ext}$.^[33] The Hom functor \operatorname{Hom}_R(M, -) applied to a short exact sequence $0 \to A \to B \to C \to 0

yields a left exact sequence &#36;0 \to \operatorname{Hom}_R(M, A) \to \operatorname{Hom}_R(M, B) \to \operatorname{Hom}_R(M, C)

, where the maps are induced by composition with i and p.^[34] This reflects the contravariant nature of \operatorname{Hom}_R(-, N), which produces $0 \to \operatorname{Hom}_R(C, N) \to \operatorname{Hom}_R(B, N) \to \operatorname{Hom}_R(A, N)for fixedN.[33] These induced sequences preserve exactness at the initial terms but may fail to be exact at \operatorname{Hom}_R(M, C)unlessM$ is projective, in which case the full sequence is exact.^[34] Short exact sequences are equivalent up to congruence if there exists an isomorphism of sequences commuting with the homomorphisms, preserving the extension class in \operatorname{Ext}^1_R(C, A).^[33] A sequence splits if there is a homomorphism s: C \to B such that p \circ s = \operatorname{id}_C, implying B \cong A \oplus C as modules; this occurs precisely when the extension class is zero.^[34] The Snake Lemma provides a long exact sequence connecting kernels and cokernels across commutative diagrams of short exact sequences, aiding diagram chasing for homomorphisms.^[33] Applying derived functors extends these to long exact sequences: for $0 \to A' \to A \to A'' \to 0, the sequence \cdots \to \operatorname{Ext}^n_R(A', B) \to \operatorname{Ext}^n_R(A, B) \to \operatorname{Ext}^n_R(A'', B) \to \operatorname{Ext}^{n+1}_R(A', B) \to \cdotsholds for any moduleB.[34] In particular, the beginning of the long exact sequence associated to the contravariant \operatorname{Hom}_R(-,B)

applied to &#36;0 \to A' \to A \to A'' \to 0

is $0 \to \operatorname{Hom}_R(A'', B) \to \operatorname{Hom}_R(A, B) \to \operatorname{Hom}_R(A', B) \to \operatorname{Ext}^1_R(A'', B) \to \cdots, where the connecting [homomorphism](/page/Homomorphism) \operatorname{Hom}_R(A', B) \to \operatorname{Ext}^1_R(A'', B)quantifies the obstructions to lifting (or extending) homomorphisms fromA'toB$ over the extension.^[33] For example, consider R = \mathbb{Z} and the sequence $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0; this is short exact but nonsplit, with \operatorname{Hom}\mathbb{Z}(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}) = 0illustrating the failure of right exactness for\operatorname{Hom}\mathbb{Z}(-, \mathbb{Z})$.^[34]

Long Exact Sequences in Homology

In homological algebra over a ring R, chain complexes provide a framework for studying module homomorphisms through sequences of modules and differentials. A chain complex C_\bullet consists of R-modules C_n for n \in \mathbb{Z} and module homomorphisms d_n: C_n \to C_{n-1} (the differentials) satisfying d_{n-1} \circ d_n = 0 for all n. The homology groups are the R-modules H_n(C_\bullet) = \ker d_n / \operatorname{im} d_{n+1}, which capture the cycles modulo boundaries. A chain map f_\bullet: A_\bullet \to B_\bullet is a family of module homomorphisms f_n: A_n \to B_n that commute with the differentials, i.e., d_n^B \circ f_n = f_{n-1} \circ d_n^A. Such maps induce well-defined module homomorphisms on homology, f_{n*}: H_n(A_\bullet) \to H_n(B_\bullet).^[11] A short exact sequence of chain complexes, $0 \to A_\bullet \xrightarrow{i_\bullet} B_\bullet \xrightarrow{p_\bullet} C_\bullet \to 0, consists of chain maps where for each degree n, the sequence $0 \to A_n \xrightarrow{i_n} B_n \xrightarrow{p_n} C_n \to 0 is a short exact sequence of R-modules. This structure induces a long exact sequence of homology modules:

\cdots \to H_n(A_\bullet) \xrightarrow{i_{n*}} H_n(B_\bullet) \xrightarrow{p_{n*}} H_n(C_\bullet) \xrightarrow{\partial_n} H_{n-1}(A_\bullet) \xrightarrow{i_{(n-1)*}} H_{n-1}(B_\bullet) \to \cdots,

extending infinitely in both directions. The connecting homomorphism \partial_n: H_n(C_\bullet) \to H_{n-1}(A_\bullet) is defined by lifting a homology class \in H_n(C_\bullet) (where c \in Z_n(C_\bullet)) to a preimage b \in B_n under p_n, noting that d_n(b) \in \operatorname{im} i_{n-1}, and then mapping to the class [d_n(b)] \in H_{n-1}(A_\bullet). This sequence is exact at each term, meaning the image of each map equals the kernel of the next.^[35]^[5]^[11] The proof relies on the snake lemma, which constructs the connecting homomorphism from commutative diagrams of short exact sequences of modules. Applied degreewise to the kernels of differentials (forming a short exact sequence of complexes) and to the images (or cokernels), the snake lemma yields exactness at the homology groups. If the original short exact sequence of complexes admits a splitting by chain maps, then the connecting homomorphisms vanish, and the induced long exact sequence splits into short exact sequences $0 \to H_n(A_\bullet) \to H_n(B_\bullet) \to H_n(C_\bullet) \to 0, each splitting as H_n(B_\bullet) \cong H_n(A_\bullet) \oplus H_n(C_\bullet). This result, fundamental since the development of [homological algebra](/page/Homological_algebra), enables computations of [homology](/page/Homology) by relating it to known cases, such as in derived functors like \operatorname{Tor}or\operatorname{Ext}, where short [exact sequence](/page/Exact_sequence)s of [module](/page/Module)s produce analogous long [exact sequence](/page/Exact_sequence)s in [homology](/page/Homology). For instance, tensoring a short [exact sequence](/page/Exact_sequence) of [module](/page/Module)s with another [module](/page/Module) may fail to be exact, but the derived functor \operatorname{Tor}$ yields a long exact sequence measuring the failure.^[35]^[5] In the category of R-modules, these long exact sequences highlight how module homomorphisms preserve or detect exactness in homological terms. They are natural in the sense that if another short exact sequence of chain maps acts on the original, the induced maps on homology commute with the long exact sequence maps. This naturality underpins applications in algebraic topology and commutative algebra, where homology modules classify invariants of modules via resolutions.^[35]^[11]

Endomorphisms

Endomorphisms of Finitely Generated Modules

In module theory, the endomorphism ring of a module M over a ring R, denoted \operatorname{End}_R(M), is the ring whose elements are all R-module homomorphisms from M to itself, equipped with pointwise addition and composition as the ring operations. For a finitely generated R-module M, where R is commutative, the finite generation implies that \operatorname{End}_R(M) is finitely presented as an R-algebra in many cases, and endomorphisms exhibit algebraic relations analogous to those in linear algebra. A fundamental property is that every endomorphism \varphi: M \to M satisfies a monic polynomial equation over R, as stated by the generalized Cayley-Hamilton theorem: there exists a monic polynomial p(x) \in R of degree equal to the number of generators of M such that p(\varphi) = 0.^[36] This result follows from representing \varphi with respect to a finite generating set and applying the classical determinant argument to the associated matrix.^[37] A notable consequence for finitely generated modules is that any surjective endomorphism is necessarily an isomorphism. Specifically, if \varphi: M \to M is surjective, then \ker(\varphi) = 0, ensuring \varphi is bijective.^[38] This holds because the finite generation allows the use of the Cayley-Hamilton theorem or Nakayama's lemma in the local case to show injectivity. For example, over a principal ideal domain (PID) R, such as \mathbb{Z} or k for a field k, finitely generated modules decompose uniquely as a direct sum of a free submodule and a torsion submodule, M \cong F \oplus T, where F is free of finite rank and T is a direct sum of cyclic torsion modules R/(d_i R) with d_i dividing d_{i+1}.^[39] The endomorphism ring then takes a block triangular form reflecting this decomposition: elements of \operatorname{End}_R(M) consist of matrices \begin{pmatrix} \alpha & \beta \\ 0 & \gamma \end{pmatrix}, where \alpha \in \operatorname{End}_R(T), \gamma \in \operatorname{End}_R(F), and \beta \in \operatorname{Hom}_R(F, T).^[40] For the free part F \cong R^n, the endomorphism ring is the matrix ring \operatorname{Mat}_n(R), consisting of n \times n matrices over R acting by left multiplication on column vectors.^[10] The torsion part's endomorphism ring is more intricate; for a cyclic torsion module R/(dR), \operatorname{End}_R(R/(dR)) \cong R/(dR), via multiplication maps. When T is a direct sum of such cyclics with coprime annihilators, the endomorphism ring decomposes as a direct product of these rings; otherwise, with chained invariant factors, it forms a ring of triangular matrices over the respective quotient rings.^[40] These structures highlight how finite generation constrains the possible endomorphisms, enabling classification and computation in applications like representation theory and algebraic geometry.

Invariants and Classification

In the context of endomorphisms of finitely generated modules over a principal ideal domain (PID), invariants play a crucial role in classifying these maps up to conjugation. For a finitely generated module M over a PID R and an endomorphism \phi: M \to M, one views M as an R-module where the action of x is given by \phi. Since R is also a PID, the structure theorem applies, decomposing M uniquely (up to isomorphism) as a direct sum of cyclic R-modules: M \cong \bigoplus_{i=1}^k R/(d_i(x)), where the d_i(x) are monic polynomials in R satisfying d_1(x) \mid d_2(x) \mid \cdots \mid d_k(x). These polynomials d_i(x) are the invariant factors of \phi and serve as complete invariants for the endomorphism up to similarity.^[41] The invariant factors determine the rational canonical form of \phi, which provides a canonical matrix representation relative to a suitable basis of M. Specifically, if \{e_{ij} \mid 1 \leq i \leq k, 1 \leq j \leq \deg d_i\} is a basis adapted to the decomposition, the matrix of \phi is block diagonal, consisting of companion matrices C(d_i(x)) for each invariant factor. The companion matrix C(d_i(x)) for a monic polynomial d_i(x) = x^{n_i} + a_{n_i-1} x^{n_i-1} + \cdots + a_0 is the n_i \times n_i matrix with last row [-a_0, -a_1, \dots, -a_{n_i-1}] and subdiagonal 1's elsewhere. Two endomorphisms are similar (i.e., conjugate via an automorphism of M) if and only if they share the same invariant factors.^[41]^[39] An alternative set of invariants is provided by the elementary divisors, which decompose each invariant factor into powers of irreducible polynomials. The structure theorem yields a decomposition M \cong \bigoplus_{j} \bigoplus_{i} R/(p_j(x)^{e_{ij}}), where p_j(x) are distinct monic irreducibles over R and e_{ij} \geq 1. These elementary divisors are unique up to ordering within each irreducible factor and also classify endomorphisms up to similarity; the rational canonical form then uses companion matrices for each p_j(x)^{e_{ij}}. For instance, when R = \mathbb{Z} (classifying endomorphisms of finitely generated abelian groups) or R = k a field (reducing to the classical rational canonical form for linear transformations), the elementary divisors correspond to the primary decomposition, with the minimal polynomial being the product of the highest powers of each irreducible.^[41]^[39] The characteristic polynomial of \phi, given by \det(xI - \phi) = \prod_{i=1}^k d_i(x), and the minimal polynomial, the least common multiple of the d_i(x), are derived from these invariants and provide partial but computable information about the structure. Over fields where irreducibles split (e.g., algebraically closed), the elementary divisors yield the Jordan canonical form as a refinement.^[39]

Variants

Additive Relations

The concept of additive relations was introduced by J. H. C. Whitehead in 1961.^[42] In the category of modules over a ring R, an additive relation from an R-module A to an R-module B is defined as a submodule \rho \subseteq A \oplus B.^[42] This structure generalizes the notion of a module homomorphism by allowing multi-valued mappings that preserve the additive and scalar multiplication operations in a set-theoretic sense. Specifically, for elements (a_1, b_1), (a_2, b_2) \in \rho and scalars r \in R, it follows that (a_1 + a_2, b_1 + b_2) \in \rho and (r a_1, r b_1) \in \rho, ensuring closure under the module operations. A module homomorphism f: A \to B corresponds precisely to its graph \Gamma_f = \{(a, f(a)) \mid a \in A\} \subseteq A \oplus B, which is an additive relation that is functional: for each a \in A, there exists exactly one b \in B such that (a, b) \in \Gamma_f. Conversely, not every additive relation is functional; for instance, the full direct sum A \oplus B itself is an additive relation, but it pairs every element of A with every element of B, representing the most permissive multi-valued map. Another example arises from submodules: if K \subseteq A and S \subseteq B, the relation \{(k, s) \mid k \in K, s \in S\} is additive if K and S are submodules, though it may not connect elements in a homomorphism-like manner unless further restricted. Additive relations form an algebra under composition and inversion. The composition \sigma \circ \rho of \rho: A \to B and \sigma: B \to C is the submodule \{(a, c) \in A \oplus C \mid \exists b \in B \text{ such that } (a, b) \in \rho \text{ and } (b, c) \in \sigma\} of A \oplus C, which is again an additive relation. The converse (or inverse) relation \rho^\#: B \to A is \{(b, a) \mid (a, b) \in \rho\} \subseteq B \oplus A, and for a homomorphism f, \Gamma_f^\# corresponds to the graph of f^{-1} when f is an isomorphism. Symmetric additive relations satisfy \rho = \rho^\#, and idempotent ones obey \rho \circ \rho = \rho; a key result is that every symmetric idempotent additive relation \rho: A \to A takes the form \rho = U_{S,K} = \{(s, s + k) \mid s \in S, k \in K\} for submodules K \subseteq S \subseteq A. These relations connect to standard homomorphisms through induced maps on subquotients. For an additive relation \rho: A \to B, one defines the defect \operatorname{Def} \rho = \{a \in A \mid \exists b \in B \text{ with } (a, b) \in \rho\} and kernel \operatorname{Ker} \rho = \{a \in \operatorname{Def} \rho \mid (a, 0) \in \rho\}, both submodules of A; similarly, the image \operatorname{Im} \rho = \{b \in B \mid \exists a \in A \text{ with } (a, b) \in \rho\} and indefect \operatorname{Ind} \rho = \{b \in \operatorname{Im} \rho \mid (0, b) \in \rho\} are submodules of B. There is then a canonical isomorphism \operatorname{Def} \rho / \operatorname{Ker} \rho \cong \operatorname{Im} \rho / \operatorname{Ind} \rho, mirroring the first isomorphism theorem for homomorphisms. When \rho is the graph of a homomorphism, \operatorname{Ker} \rho = \ker f, \operatorname{Ind} \rho = 0, and the isomorphism recovers \operatorname{im} f \cong A / \ker f. Applications of additive relations extend beyond direct generalizations of homomorphisms. In homological algebra, they naturally encode secondary cohomology operations, which are defined on cohomology classes in the cokernels of primary operations and map to other cohomology groups via such relations rather than single-valued functions. For example, if primary operations have kernels as submodules, secondary operations arise as additive relations between those kernels. They also facilitate definitions of connecting homomorphisms in exact sequences of modules, where the connecting map can be viewed as an additive relation linking the cokernel of one map to the kernel of the next. This framework applies more broadly to any abelian category, including the category of R-modules, providing a unified treatment of multi-valued structures in algebraic topology and homological contexts.

Bilinear Maps as Variants

In the context of module theory, a bilinear map provides a natural extension of the concept of a module homomorphism by incorporating linearity in two separate arguments. Specifically, given an associative ring R with identity and left R-modules M, N, and P, an R-bilinear map \phi: M \times N \to P is a function that is R-linear in each variable when the other is fixed: for all m, m' \in M, n, n' \in N, p \in P, and r \in R,

\phi(rm + m', n) = r\phi(m, n) + \phi(m', n), \quad \phi(m, rn + n') = r\phi(m, n) + \phi(m, n'),

and \phi(m + m', n) = \phi(m, n) + \phi(m', n), \phi(m, n + n') = \phi(m, n) + \phi(m, n').^[43] This structure captures interactions between two modules in a way that parallels the single-argument linearity of standard module homomorphisms.^[44] The connection to module homomorphisms arises through the tensor product M \otimes_R N, which serves as a universal object linearizing bilinear maps. For any R-bilinear map \phi: M \times N \to P, there exists a unique R-module homomorphism \tilde{\phi}: M \otimes_R N \to P such that \phi(m, n) = \tilde{\phi}(m \otimes n) for all m \in M, n \in N. Conversely, every R-module homomorphism f: M \otimes_R N \to P induces an R-bilinear map via \phi(m, n) = f(m \otimes n). This bijection establishes bilinear maps as equivalent to homomorphisms from the tensor product, effectively reducing multi-variable linearity to single-variable cases.^[43]^[45] Equivalently, fixing the second argument yields another perspective: the set of R-bilinear maps M \times N \to P is in natural bijection with the set of R-module homomorphisms M \to \mathrm{Hom}_R(N, P), where \mathrm{Hom}_R(N, P) denotes the R-module of homomorphisms from N to P. For a bilinear \phi, the corresponding homomorphism sends m \mapsto \phi_m, where \phi_m(n) = \phi(m, n) is linear in n. This adjunction highlights bilinear maps as a "curried" variant of homomorphisms, transforming joint linearity into iterated single linearity.^[44]^[46] A canonical example is the multiplication map in an R-algebra A, viewed as modules over R: the map \mu: A \times A \to A given by (a, b) \mapsto ab is R-bilinear, inducing the homomorphism A \otimes_R A \to A via a \otimes b \mapsto ab. This illustrates how bilinear maps encode algebraic structures like products, generalizing the role of homomorphisms in preserving operations.^[43] In noncommutative settings or over noncommutative rings, additional care is needed for right/left module distinctions, but the core correspondence persists.^[47]

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