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Direct sum of modules

In module theory, the direct sum of a family of modules \{M_i\}_{i \in I} over a ring R is the R-module consisting of all families (m_i)_{i \in I} with m_i \in M_i and all but finitely many m_i = 0, equipped with componentwise addition and . For finite families, such as two modules M and N, the M \oplus N comprises all pairs (m, n) without the finiteness restriction, and it coincides with the in this case. This construction serves as the in the of R-modules, characterized by the universal property that for any module L with homomorphisms \phi_i: M_i \to L, there exists a unique \Phi: \bigoplus_i M_i \to L such that \Phi \circ \iota_i = \phi_i for maps \iota_i. An internal direct sum arises within a single M: if M_1, \dots, M_n are submodules such that M = M_1 + \dots + M_n and M_i \cap (\sum_{j \neq i} M_j) = \{0\} for each i, then M \cong M_1 \oplus \dots \oplus M_n, with every element of M admitting a unique decomposition as a sum of elements from the submodules. A submodule S \subseteq M is a direct summand if it is complemented by another submodule T such that M \cong S \oplus T. For infinite families, the embeds as a submodule into the \prod_i M_i, which allows arbitrary families without the finiteness condition, highlighting a key distinction in infinite cases. Direct sums play a central role in the structure theory of modules, particularly over principal ideal domains (PIDs), where every finitely generated decomposes uniquely (up to ) as a of cyclic modules, enabling the of such modules via factors or elementary divisors. They also facilitate the study of properties like projectivity and injectivity, as of projective (resp., injective) modules are projective (resp., injective), though not all properties preserve under infinite sums. This decomposition tool extends concepts from vector spaces and abelian groups, underscoring the 's foundational importance in .

Constructions in Basic Categories

Vector Spaces over a Field

In the category of vector spaces over a K, the direct sum of two spaces V and W is constructed as the set V \oplus W consisting of all ordered pairs (v, w) with v \in V and w \in W, equipped with componentwise addition (v, w) + (v', w') = (v + v', w + w') and \lambda (v, w) = (\lambda v, \lambda w) for \lambda \in K. This structure makes V \oplus W a over K, serving as the categorical , where morphisms from V and W combine uniquely into a morphism to another space. The basis of V \oplus W is formed by the of bases for V and W; if \{v_i\} is a basis for V and \{w_j\} is a basis for W, then \{(v_i, 0) \mid i\} \cup \{(0, w_j) \mid j\} spans V \oplus W and is linearly independent. Consequently, for finite-dimensional spaces, the satisfies \dim(V \oplus W) = \dim V + \dim W. For instance, the \mathbb{R}^2 \oplus \mathbb{R}^3 is isomorphic to \mathbb{R}^5, where elements are represented as 5-tuples (x_1, x_2, x_3, x_4, x_5) corresponding to ((x_1, x_2), (x_3, x_4, x_5)), with the of \mathbb{R}^5 arising from the of the bases of \mathbb{R}^2 and \mathbb{R}^3. The natural inclusion maps are defined by \iota_V: V \to V \oplus W, \iota_V(v) = (v, 0), and \iota_W: W \to V \oplus W, \iota_W(w) = (0, w); these are linear injections whose images intersect trivially and span V \oplus W. Historically, for finite collections of vector spaces, this direct sum coincides with the Cartesian product, as every element involves only finitely many components, eliminating the distinction that arises in infinite cases.

Abelian Groups

In the category of , the direct sum G \oplus H of two G and H is defined as the set of ordered pairs (g, h) with g \in G and h \in H, equipped with componentwise addition (g_1, h_1) + (g_2, h_2) = (g_1 + g_2, h_1 + h_2). This construction serves as the in the category of , meaning that for any K and group s f: G \to K, g: H \to K, there exists a unique \phi: G \oplus H \to K such that \phi \circ i_G = f and \phi \circ i_H = g, where i_G: G \to G \oplus H and i_H: H \to G \oplus H are the inclusion maps given by i_G(g) = (g, 0) and i_H(h) = (0, h). A basic example is the direct sum \mathbb{Z} \oplus \mathbb{Z}, which is isomorphic to \mathbb{Z}^2, the free abelian group of rank 2 generated by the standard basis elements (1,0) and (0,1). For finite cyclic groups with coprime orders, the direct sum simplifies further; for instance, \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} \cong \mathbb{Z}/6\mathbb{Z} via the Chinese Remainder Theorem, where the isomorphism maps (1,1) to the generator of \mathbb{Z}/6\mathbb{Z}. The fundamental theorem of finitely generated abelian groups illustrates the role of direct sums in decomposition: every finitely generated abelian group is isomorphic to a direct sum of a free abelian group (its torsion-free part) and its torsion subgroup, which decomposes as a direct sum of its p-primary components for each prime p. For example, the torsion subgroup of a finitely generated abelian group is a direct sum of cyclic groups of prime-power order, such as \mathbb{Z}/p^k\mathbb{Z} for distinct primes p. The projection maps from the direct sum are defined componentwise: the projection \pi_G: G \oplus H \to G sends (g, h) \mapsto g, and similarly \pi_H: G \oplus H \to H sends (g, h) \mapsto h. These projections are homomorphisms that are compatible with the coproduct structure, as any homomorphism out of G \oplus H factors uniquely through the projections when composed with the inclusions. For finite direct sums of abelian groups, the direct sum coincides with the , as every element has components in only finitely many factors. However, for infinite families, the restricts to elements with only finitely many nonzero components, distinguishing it from the full .

General Constructions for Modules

Direct Sum of Two Modules

In , the of two modules M and N over a R with , denoted M \oplus N, is defined as the set \{(m, n) \mid m \in M, n \in N\} equipped with componentwise addition (m_1, n_1) + (m_2, n_2) = (m_1 + m_2, n_1 + n_2) and scalar multiplication r \cdot (m, n) = (r \cdot m, r \cdot n) for r \in R. This structure makes M \oplus N an R-module, where the R acts diagonally on the pairs. The M \oplus N is characterized by the universal mapping property: for any R- P and any R- homomorphisms f: M \to P, g: N \to P, there exists a unique R- h: M \oplus N \to P such that h(m, n) = f(m) + g(n). This homomorphism h is determined by the inclusion maps \iota_M: M \to M \oplus N given by \iota_M(m) = (m, 0) and \iota_N: N \to M \oplus N given by \iota_N(n) = (0, n), satisfying h \circ \iota_M = f and h \circ \iota_N = g. When R = \mathbb{Z}, the direct sum of two abelian groups recovers the standard , as abelian groups are precisely the \mathbb{Z}-modules. For non-commutative rings, such as the ring of n \times n matrices over a , the direct sum construction applies similarly, with modules over matrix rings corresponding to direct sums of copies of column vector spaces. This construction for two modules addresses finite direct sums of exactly two terms and extends to finite sums of more modules through iterated application of the binary operation.

Direct Sum of an Arbitrary Family

In the context of an arbitrary I and a family of R-modules \{M_i\}_{i \in I}, the \bigoplus_{i \in I} M_i is defined as the set of all families (m_i)_{i \in I} where m_i \in M_i for each i \in I and m_i = 0 for all but finitely many i, equipped with componentwise addition and : (m_i) + (m_i') = (m_i + m_i') and r \cdot (m_i) = (r m_i) for r \in R. This finite support condition ensures that the direct sum captures only "finitely generated" combinations from the family, distinguishing it from other constructions. The can be constructed explicitly as a submodule of the \prod_{i \in I} M_i, which consists of all families without the finite support restriction; specifically, \bigoplus_{i \in I} M_i is the submodule generated by the elements, where for each i \in I and m \in M_i, the element e_{i,m} has m in the i-th component and 0 elsewhere. The inclusion maps \iota_i: M_i \to \bigoplus_{j \in I} M_j are given by \iota_i(m) = (\delta_{ij} m)_{j \in I}, where \delta_{ij} is the , embedding each M_i as the submodule with support only at i. A representative example is the \bigoplus_{n=1}^\infty \mathbb{Z}, which is isomorphic to the on countably infinitely many generators, consisting of all linear combinations with only finitely many nonzero coefficients. In contrast, the \prod_{n=1}^\infty \mathbb{Z} allows arbitrary sequences and has $2^{\aleph_0}, making it uncountable and non-free. By convention, the over an empty is the zero , the unique with a single element satisfying the . This aligns with the finite case, where the of no modules yields the trivial structure.

Properties

Algebraic Properties

The of two M and N over a R is equipped with componentwise and , defined by (m_1, n_1) + (m_2, n_2) = (m_1 + m_2, n_1 + n_2) and r(m, n) = (rm, rn) for r \in R, m \in M, and n \in N. This structure ensures additivity in each component, as (m_1 + m_2, n) = (m_1, n) + (m_2, n), and bilinearity with respect to the ring action, preserving the axioms. The operation of forming direct sums is commutative and associative up to canonical isomorphisms: there exists an M \oplus N \cong N \oplus M given by (m, n) \mapsto (n, m), and (M \oplus N) \oplus P \cong M \oplus (N \oplus P) via the map ((m, n), p) \mapsto (m, (n, p)). Additionally, the direct sum with the zero module satisfies M \oplus 0 \cong M through the explicit (m, 0) \leftrightarrow m. For finite families of modules \{M_i\}_{i=1}^k, the \bigoplus_{i=1}^k M_i coincides with the iterated pairwise direct sum, inheriting these properties inductively. A key homological property is the \Hom_R\left(\bigoplus_{i \in I} M_i, N\right) \cong \prod_{i \in I} \Hom_R(M_i, N), where a from the to N is uniquely determined by its restrictions to each summand, which act independently. This reflects the nature of direct sums in the category of . Furthermore, direct sums preserve projectivity: if each M_i is a projective R-, then \bigoplus_{i \in I} M_i is projective, as it arises as a direct summand of a .

Universal Property

In the category of left R-modules, denoted \mathrm{Mod}_R, the of two s M and N satisfies a characterizing it as the . Specifically, for any R- P and any pair of R-s f: M \to P and g: N \to P, there exists a unique R- h: M \oplus N \to P such that the following diagrams commute: h \circ \iota_M = f and h \circ \iota_N = g, where \iota_M: M \to M \oplus N and \iota_N: N \to M \oplus N are the inclusion maps sending m \mapsto (m, 0) and n \mapsto (0, n), respectively. This can be visualized as a commutative square where M and N map to P directly via f and g, and also via the inclusions to M \oplus N followed by h. The uniqueness of h ensures that M \oplus N, together with the inclusions, is initial among all objects receiving such a pair of maps from M and N. This property extends to the direct sum of an arbitrary family of modules \{M_i\}_{i \in I} over an index set I, which may be infinite. The object \bigoplus_{i \in I} M_i comes equipped with canonical inclusions \iota_i: M_i \to \bigoplus_{i \in I} M_i for each i \in I. For any R-module P and any family of R-module homomorphisms \{f_i: M_i \to P\}_{i \in I}, there exists a unique R-module homomorphism h: \bigoplus_{i \in I} M_i \to P such that h \circ \iota_i = f_i for all i \in I. In diagrammatic terms, this forms a commutative cone: each M_i maps to P via f_i, and equivalently via \iota_i to the direct sum followed by h, with the universal morphism h factoring uniquely through any such family. The finite case is recovered when I has two elements. To see why this holds, consider the explicit construction of h. An element of \bigoplus_{i \in I} M_i is a tuple (m_i)_{i \in I} with only finitely many nonzero m_i, so define h((m_i)_{i \in I}) = \sum_{i \in I} f_i(m_i); this is well-defined because the sum is finite and the f_i are linear. Linearity of h follows from linearity of the f_i, and the relation h \circ \iota_i = f_i holds by direct computation on generators. Uniqueness arises because any such h is determined on the images of the \iota_i, which generate the direct sum as an R-module. In the category \mathrm{Mod}_R, the is thus given by the , a feature shared with the category of abelian groups () and vector spaces over a (Vect_k). This contrasts with non-abelian , such as the category of groups, where coproducts take the form of products rather than direct sums.

Comparison with Direct Product

In the case of a finite , the and of modules coincide up to canonical . For modules M_1, \dots, M_n over a R, the \bigoplus_{i=1}^n M_i is isomorphic to the \prod_{i=1}^n M_i as R-modules, where both are realized as the set of n-tuples (m_1, \dots, m_n) with m_i \in M_i and componentwise addition and .$$] When the index set I is infinite, the direct sum and direct product diverge set-theoretically and algebraically. The direct product \prod_{i \in I} M_i comprises all families (m_i)_{i \in I} with m_i \in M_i for each i, again with componentwise operations. The direct sum \bigoplus_{i \in I} M_i, however, consists precisely of those families where m_i = 0 for all but finitely many i, forming a proper submodule of the direct product whenever the M_i are nonzero.[ [22][24] The natural inclusion $\bigoplus_{i \in I} M_i \hookrightarrow \prod_{i \in I} M_i$ is injective but not surjective in this setting.] This distinction is evident in the category of abelian groups. The direct sum \bigoplus_{n \in \mathbb{N}} \mathbb{Z} is free on a countable basis \{e_n \mid n \in \mathbb{N}\}, where e_n has 1 in the nth position and 0 elsewhere. In contrast, the direct product \prod_{n \in \mathbb{N}} \mathbb{Z} consists of all integer sequences and is uncountable, with cardinality $2^{\aleph_0}. Considering the product as a ring under componentwise multiplication, the quotient by the sum submodule admits zero divisors, such as the images of characteristic functions of disjoint infinite subsets of \mathbb{N}, whose product is zero.[$$ The constructions also differ in their interactions with the Hom functor. For any R-module N, the universal property of the coproduct yields \mathrm{Hom}_R\left( \bigoplus_{i \in I} M_i, N \right) \cong \prod_{i \in I} \mathrm{Hom}_R(M_i, N). Likewise, the universal property of the product gives \mathrm{Hom}_R\left( N, \prod_{i \in I} M_i \right) \cong \prod_{i \in I} \mathrm{Hom}_R(N, M_i). There is a natural injection \bigoplus_{i \in I} \mathrm{Hom}_R(N, M_i) \hookrightarrow \mathrm{Hom}_R\left( N, \bigoplus_{i \in I} M_i \right), corresponding to maps that factor through finite direct sums, but in general this is not surjective for infinite I. The isomorphism holds when I is finite. For maps out of the product, there is a natural injection \bigoplus_{i \in I} \mathrm{Hom}_R(M_i, N) \hookrightarrow \mathrm{Hom}_R\left( \prod_{i \in I} M_i, N \right). In general, \mathrm{Hom}_R\left( \prod_{i \in I} M_i, N \right) properly contains \bigoplus_{i \in I} \mathrm{Hom}_R(M_i, N), with equality holding under additional conditions on N, such as N being slender when the M_i are cyclic. Over a R, both the and serve as biproducts in the of R-modules precisely when the is finite.$$]

Internal Direct Sum

Definition and Criteria

In module theory, given an R-module M and a of submodules \{N_i\}_{i \in I} of M, the submodules form an internal direct sum if every element m \in M can be uniquely written as a finite sum m = \sum_{i \in F} n_i, where F \subset I is finite and each n_i \in N_i. This uniqueness ensures that the representation is independent of choices within the submodules. A necessary and sufficient criterion for the internal is that the of the submodules equals the entire , \sum_{i \in I} N_i = M, and that for each i \in I, the N_i \cap \sum_{j \neq i} N_j = \{0\}. For a finite family of two submodules N, P \subseteq M, this simplifies to M = N + P and N \cap P = \{0\}, in which case M is denoted M = N \oplus P. In this finite case, the condition can equivalently be checked pairwise for the intersections with the sums excluding each submodule. For instance, the \mathbb{Z}^2 (as a \mathbb{Z}-) is the internal of the submodules N = \langle (1,0) \rangle = \mathbb{Z} \times \{0\} and P = \langle (0,1) \rangle = \{0\} \times \mathbb{Z}, since every (a,b) \in \mathbb{Z}^2 uniquely as (a,0) + (0,b) with (a,0) \in N, (0,b) \in P, N + P = \mathbb{Z}^2, and N \cap P = \{0\}. In contrast, the rational numbers \mathbb{Q} as a \mathbb{Z}- provide a non-example: it admits no decomposition into an internal of two nonzero proper submodules, as any pair of nonzero submodules N, P \subseteq \mathbb{Q} with N + P = \mathbb{Q} necessarily satisfies N \cap P \neq \{0\}. If the family \{N_i\} forms an internal direct sum of M, then the canonical sum map \sum_{i \in I} \iota_i : \bigoplus_{i \in I} N_i \to M, where each \iota_i : N_i \to M is the , is an of R-modules. This concept of internal direct sum generalizes the classical direct sum decompositions of vector spaces over a into subspaces spanned by basis elements.

Relation to Submodule Decompositions

The internal direct sum decomposition of a module M into submodules N_i implies that M is isomorphic to the external direct sum \bigoplus N_i, where the isomorphism arises from the inclusion maps and the fact that the summands intersect trivially and generate M. This equivalence holds because the internal construction satisfies the universal property of the in the category of modules, ensuring a canonical between elements. Free modules admit explicit decompositions as internal direct sums of rank-one free modules. Specifically, a free module of finite rank n over a ring R is isomorphic to R^n = \bigoplus_{i=1}^n R, where each R is a rank-one free module generated by a basis element. Projective modules generalize this by being direct summands of free modules; thus, every projective module P decomposes as an internal direct summand in some free module F = P \oplus Q. The Krull–Schmidt theorem provides conditions for unique decompositions. For a M of finite over any , M decomposes as a of indecomposable modules, and any two such decompositions are unique up to and reordering of the summands. This uniqueness follows from the structure of endomorphisms on indecomposables, where non-isomorphism implies zero maps between distinct summands. Indecomposable modules illustrate basic obstructions to non-trivial decompositions, as they admit no internal M = N \oplus K with both N and K non-zero. In the category of abelian groups (i.e., \mathbb{Z}-modules), the fundamental theorem of finitely generated abelian groups guarantees a unique into a torsion submodule (direct sum of cyclic groups of order) and a torsion-free part (free abelian of finite rank). However, not all modules decompose non-trivially. For example, \mathbb{Q} as a \mathbb{Z}-module is indecomposable, possessing no non-trivial direct summands despite being torsion-free and divisible. In cases where decompositions exist, uniqueness often stems from orthogonal projections in the endomorphism ring \operatorname{End}(M): for summands N_i, there exist idempotents e_i (satisfying e_i^2 = e_i) such that e_i e_j = 0 for i \neq j and \sum e_i = \operatorname{id}_M, with \operatorname{im}(e_i) = N_i. These projections ensure the summands are canonically determined.

Grothendieck Construction

The Grothendieck Group

The Grothendieck group K_0(R) of a R is defined as the generated by the isomorphism classes of R-, denoted [M] for an R- M, subject to the relations [M] + [N] = [M \oplus N] for all M and N. This construction can be formalized as K_0(R) = \mathbb{Z}^{( \mathrm{Iso}(R\text{-Mod}) )} / \sim, where \mathbb{Z}^{(S)} denotes the on a set S, and the \sim identifies classes via the operation, incorporating formal differences [M] - [N] to embed the commutative ( \mathrm{Iso}(R\text{-Mod}), \oplus, 0 ) into an . More precisely, it arises as the universal making the map from the of isomorphism classes under to the group a homomorphism, ensuring additivity with respect to : [M \oplus N] = [M] + [N]. This group completion captures relations induced by short exact sequences in the category of R-modules. Specifically, for a short exact sequence $0 \to A \to B \to C \to 0, the classes satisfy [B] = [A] + [C] in K_0(R), reflecting the in . In the standard presentation for algebraic , K_0(R) is generated by classes [P] of finitely generated projective R-modules, with relations arising from projective resolutions or split exact sequences involving such modules. For the ring R = \mathbb{Z}, the Grothendieck group K_0(\mathbb{Z}) is isomorphic to \mathbb{Z}, where the isomorphism is given by the rank function on free modules, as all finitely generated projective \mathbb{Z}-modules are free. Similarly, for R a field, K_0(R) \cong \mathbb{Z}, again via the dimension (rank) of vector spaces, since all modules over a field are free (or zero). The concept was introduced by in the late 1950s as a foundational tool in algebraic , initially to generalize the Riemann-Roch theorem in the context of coherent sheaves on algebraic varieties, later extended to modules over .

Direct Sums in the

In the K_0(R) of a R, the operation on modules induces the additive group structure through the [\cdot] : \mathrm{Iso}(R\text{-Mod}) \to K_0(R), which sends isomorphism classes of modules to their classes in the group and acts as a homomorphism with respect to \oplus. Specifically, for any modules M and N, the relation [M \oplus N] = [M] + [N] holds, preserving the abelian structure of isomorphism classes under . This additivity ensures that K_0(R) captures the formal differences of module classes, with the providing the underlying operation that extends to the . For a finite family of modules \{M_i\}_{i=1}^n, the class of their satisfies [ \left[ \bigoplus_{i=1}^n M_i \right] = \sum_{i=1}^n [M_i] in $ K_0(R) $, reflecting the bilinearity of the construction.[](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf) Moreover, $ K_0(R) $ is generated as an [abelian group](/page/Abelian_group) by the classes of finitely generated projective modules under these direct sums, since every element can be expressed as $ [P] - [Q] $ for projectives $ P $ and $ Q $, with free modules $ R^k $ forming a cofinal [subset](/page/Subset).[](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf) This generation property highlights how direct sums of projectives underpin the entire structure of $ K_0(R) $. The additivity extends to exact sequences: for a short exact sequence $ 0 \to A \to B \to C \to 0 $, if the sequence splits, then $ B \cong A \oplus C $ and thus $ [B] = [A] + [C] $.[](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf) In general, the [Grothendieck group](/page/Grothendieck_group) incorporates the relation $ [B] = [A] + [C] $ for any such sequence, defining the [Euler characteristic](/page/Euler_characteristic) $ \chi = [A] - [B] + [C] = 0 $, which generalizes additivity beyond split cases and links to homological invariants.[](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf) A concrete example arises over a [field](/page/Field) $ k $, where $ K_0(k) \cong \mathbb{Z} $ and the [dimension](/page/Dimension) function $ \dim : K_0(k) \to \mathbb{Z} $ given by $ [V] \mapsto \dim_k(V) $ is a [group homomorphism](/page/Group_homomorphism) additive over direct sums, since $ \dim_k(V \oplus W) = \dim_k(V) + \dim_k(W) $.[](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf) All finitely generated projective $ k $-modules are [free](/page/Free), so classes are multiples of $ $, with $ [k^n] = n \cdot $. While formal direct sums of infinitely many modules appear in $ K_0(R) $ as infinite formal sums of classes, the construction focuses on finite direct sums to ensure well-definedness in the group; infinite cases require additional [topology](/page/Topology) for [convergence](/page/Convergence) but are not central to the standard [algebraic structure](/page/Algebraic_structure).[](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf) ## Direct Sums with Extra Structure ### Direct Sum of Algebras The direct sum of a family of $R$-algebras $\{A_i\}_{i \in I}$, where $R$ is a [commutative ring](/page/Commutative_ring), is defined on the underlying $R$-module direct sum $\bigoplus_{i \in I} A_i$, consisting of tuples $(a_i)_{i \in I}$ with $a_i \in A_i$ and $a_i = 0$ for all but finitely many $i$. Addition and [scalar multiplication](/page/Scalar_multiplication) by elements of $R$ are performed componentwise: $(a_i) + (b_i) = (a_i + b_i)$ and $r \cdot (a_i) = (r a_i)$. The [multiplication](/page/Multiplication) is also componentwise: $(a_i)(b_i) = (a_i b_i)$, where $a_i b_i$ denotes the product in $A_i$. For finite index sets $I$, the unit element is the tuple $(1_i)_{i \in I}$, where $1_i$ is the multiplicative identity in $A_i$. For infinite $I$, the [direct product](/page/Direct_product) $\prod_{i \in I} A_i$ (allowing arbitrary support) is typically used instead to obtain a unital algebra, with the same componentwise operations. This structure makes $\bigoplus_{i \in I} A_i$ (or the product for infinite $I$) into an $R$-algebra, with the inclusions $\iota_i: A_i \to \bigoplus_{i \in I} A_i$ given by $\iota_i(a) = (0, \dots, a, \dots, 0)$ (with $a$ in the $i$-th position) being $R$-algebra homomorphisms that preserve multiplication: $\iota_i(a b) = \iota_i(a) \iota_i(b)$.[](https://www.cip.ifi.lmu.de/~grinberg/t/23wa/lec22.pdf)[](http://www2.math.ou.edu/~kmartin/quaint/ch2.pdf) The [direct sum](/page/Direct_sum) contains a [family](/page/Family) of orthogonal idempotents $\{e_i\}_{i \in I}$, where $e_i = \iota_i(1_i)$, satisfying $e_i^2 = e_i$, $e_i e_j = 0$ for $i \neq j$. For finite $I$, $\sum_{i \in I} e_i = (1_i)_{i \in I}$, [the unit](/page/The_Unit). These idempotents project onto the $i$-th component: $e_i ((a_j)) = ( \delta_{ij} a_i )$.[](https://www.math.uci.edu/~brusso/BremnerEtAl35pp.pdf) A concrete example is the [direct sum](/page/Direct_sum) $\mathbb{C} \oplus \mathbb{R}$ as $\mathbb{R}$-[algebra](/page/Algebra)s, where $\mathbb{C}$ is viewed as a 2-dimensional $\mathbb{R}$-[algebra](/page/Algebra) via its [standard basis](/page/Standard_basis) $\{1, i\}$ and $\mathbb{R}$ as the 1-dimensional $\mathbb{R}$-[algebra](/page/Algebra). The resulting structure is a 3-dimensional commutative $\mathbb{R}$-[algebra](/page/Algebra) with componentwise [multiplication](/page/Multiplication), such as $(c, r)(c', r') = (c c', r r')$ for $c, c' \in \mathbb{C}$ and $r, r' \in \mathbb{R}$, and [unit](/page/Unit) $(1, 1)$. It admits zero divisors, for instance $(1, 0)(0, 1) = (0, 0)$, and the idempotents are $e_1 = (1, 0)$ and $e_2 = (0, 1)$. Topologically, $\mathbb{C} \oplus \mathbb{R}$ resembles $\mathbb{C} \times \mathbb{R}$, but algebraically it is the [direct sum](/page/Direct_sum) with the specified operations.[](http://www2.math.ou.edu/~kmartin/quaint/ch2.pdf) In contrast to the tensor product, which serves as the coproduct in the category of commutative $R$-algebras, the direct sum (for finite families) functions as the categorical product in the [category](/page/Category) of $R$-algebras (commutative or not).[](https://www.cip.ifi.lmu.de/~grinberg/t/23wa/lec22.pdf) ### Direct Sum of Banach Spaces The direct sum of a family of Banach spaces $\{X_i\}_{i \in I}$ over $\mathbb{R}$ or $\mathbb{C}$ is the set $\bigoplus_{i \in I} X_i$ consisting of all families $(x_i)_{i \in I}$ where $x_i \in X_i$ and only finitely many $x_i$ are nonzero, equipped with componentwise addition and [scalar multiplication](/page/Scalar_multiplication).[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf) This construction extends the algebraic direct sum of modules to the category of normed spaces.[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf) To endow $\bigoplus_{i \in I} X_i$ with a norm, common choices include the $\ell^1$ norm $\|(x_i)\| = \sum_{i \in I} \|x_i\|_{X_i}$ or the max norm $\|(x_i)\| = \sup_{i \in I} \|x_i\|_{X_i}$.[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf)[](https://www.math.cuhk.edu.hk/course_builder/2324/math4010/Fumctional%20Analysis%202023-24Nov24.pdf) For finite index sets $I$, both norms yield a [Banach space](/page/Banach_space) whenever each $X_i$ is Banach, as Cauchy sequences converge componentwise in each coordinate.[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf) For infinite $I$, the space $\bigoplus_{i \in I} X_i$ equipped with the $\ell^1$ norm is incomplete; its [completion](/page/Completion) consists of all $(x_i)$ such that $\sum_{i \in I} \|x_i\|_{X_i} < \infty$, and this $\ell^1$-direct [sum](/page/Sum) is a [Banach space](/page/Banach_space).[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf) Similarly, the [completion](/page/Completion) under the max norm comprises all $(x_i)$ with $\sup_{i \in I} \|x_i\|_{X_i} < \infty$, forming a [Banach space](/page/Banach_space).[](https://www.math.cuhk.edu.hk/course_builder/2324/math4010/Fumctional%20Analysis%202023-24Nov24.pdf) A representative example is the space $\ell^1(\mathbb{N})$, which arises as the $\ell^1$-direct sum (completion) of countably many copies of $\mathbb{C}$, where sequences have finite $\sum |z_n|$.[](https://physics.bme.hu/sites/physics.bme.hu/files/users/BMETE15AF53_kov/Kreyszig%20-%20Introductory%20Functional%20Analysis%20with%20Applications%20%281%29.pdf) In contrast, the space $c_0$ of sequences converging to zero under the sup norm is not obtained as such a direct sum completion for copies of $\mathbb{C}$, as the completion of finite-support sequences under the max norm yields $\ell^\infty$ instead.[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf) Bounded linear operators on direct sums are defined componentwise: for families of bounded operators $T_i: X_i \to Y_i$, the direct sum $\bigoplus_{i \in I} T_i$ acts on $\bigoplus_{i \in I} X_i$ by $(\bigoplus T_i)(x_j) = (T_j x_j)_j$, preserving finite support.[](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf) The operator norm of $\bigoplus T_i$ under the $\ell^1$ or max norm on the [domain](/page/Domain) and [codomain](/page/Codomain) is $\sup_{i \in I} \|T_i\|$, ensuring boundedness if each $T_i$ is bounded.[](https://people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf) The natural inclusions $\iota_i: X_i \hookrightarrow \bigoplus_{j \in I} X_j$ are defined by $\iota_i(x) = (\delta_{ij} x)_j$, where $\delta_{ij}$ is the [Kronecker delta](/page/Kronecker_delta); these are bounded linear maps with $\|\iota_i(x)\| = \|x\|_{X_i}$ for the $\ell^1$ norm (or adjusted for finite support).[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf) Convergence in the direct sum requires that sequences of finite-support elements have coordinates converging in each $X_i$, with the sum of norms controlled for $\ell^1$-type limits.[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf) More generally, $\ell^p$-direct sums for $1 \leq p < \infty$ equip $\bigoplus_{i \in I} X_i$ with $\|(x_i)\|_p = \left( \sum_{i \in I} \|x_i\|_{X_i}^p \right)^{1/p}$, and the [completion](/page/Completion)—comprising $(x_i)$ with $\sum \|x_i\|^p < \infty$—is a [Banach space](/page/Banach_space).[](https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf) These variants generalize the scalar case, where $\ell^p(\mathbb{N})$ emerges as the [completion](/page/Completion).[](https://physics.bme.hu/sites/physics.bme.hu/files/users/BMETE15AF53_kov/Kreyszig%20-%20Introductory%20Functional%20Analysis%20with%20Applications%20%281%29.pdf) ### Direct Sum of Modules with Bilinear Forms In the context of module theory over a [commutative ring](/page/Commutative_ring) $R$, the direct sum of modules equipped with bilinear forms extends the standard construction by inducing a compatible bilinear structure on the sum. Suppose $\{M_i\}_{i \in I}$ is a [family](/page/Family) of $R$-modules, each endowed with a bilinear form $B_i: M_i \times M_i \to R$. The direct sum [module](/page/Module) $M = \bigoplus_{i \in I} M_i$ is equipped with the bilinear form $B: M \times M \to R$ defined by $B((m_i)_{i \in I}, (n_i)_{i \in I}) = \sum_{i \in I} B_i(m_i, n_i)$, where the sum is finite since only finitely many components are nonzero in each argument.[](https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf) This form ensures orthogonality between distinct components: if $\iota_i: M_i \to M$ denotes the inclusion map, then cross terms vanish, meaning $B(\iota_i(m), \iota_j(n)) = 0$ for $i \neq j$.[](https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf) More precisely, the induced form satisfies $B(\iota_i(m), \iota_j(n)) = \delta_{ij} B_i(m, n)$, where $\delta_{ij}$ is the [Kronecker delta](/page/Kronecker_delta). This orthogonality implies that the adjoint maps associated to the $B_i$ decompose accordingly on the [direct sum](/page/Direct_sum). A key preservation property holds for non-degeneracy: assuming each $M_i$ is finitely generated projective and each $B_i$ is nonsingular (i.e., the adjoint $M_i \to \mathrm{Hom}_R(M_i, R)$ is an [isomorphism](/page/Isomorphism)), then $B$ is nonsingular on $M$. Conversely, the orthogonal [direct sum](/page/Direct_sum) of nonsingular forms yields a nonsingular form.[](https://core.ac.uk/download/pdf/197757827.pdf) Examples illustrate this construction effectively. For quadratic forms, which arise from symmetric bilinear forms via [polarization](/page/Polarization), the [direct sum](/page/Direct_sum) on free modules $R^n \oplus R^m$ inherits a quadratic form that decomposes as the [sum](/page/Sum) of the [individual](/page/Individual) [ones](/page/The_Ones), facilitating [classification](/page/Classification) over rings like Dedekind domains. [Symplectic](/page/Symplectic) [direct sums](/page/Direct_sum) involve alternating bilinear forms, where the total form remains alternating and non-degenerate if each component is, as seen in the [decomposition](/page/Decomposition) of symplectic modules into hyperbolic planes. Orthogonal direct sums, for symmetric forms, similarly preserve the [signature](/page/Signature) or [discriminant](/page/Discriminant) in appropriate settings.[](https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf)[](https://core.ac.uk/download/pdf/197757827.pdf) In [representation theory](/page/Representation_theory), this structure is crucial for decomposing modules with [invariant](/page/Invariant) bilinear forms. For a group $G$-module $V$ admitting a $G$-[invariant](/page/Invariant) bilinear form $B$, if $V$ decomposes as a [direct sum](/page/Direct_sum) of [invariant](/page/Invariant) submodules $V = \bigoplus V_k$, the form restricts to each $V_k$ and induces orthogonal components, enabling the analysis of irreducible representations via [Schur's lemma](/page/Schur's_lemma), where [invariant](/page/Invariant) forms are unique up to scalar and non-degenerate. Such decompositions underpin the study of orthogonal and [symplectic](/page/Symplectic) representations, classifying [invariant](/page/Invariant) forms by [character](/page/Character) values like $\sum \chi(s^2)/|G|$.[](https://math.berkeley.edu/~serganov/math252/notes5.pdf) ### Direct Sum of Hilbert Spaces The orthogonal direct sum of a family of Hilbert spaces $\{H_i\}_{i \in I}$, denoted $\bigoplus_{i \in I} H_i$, is defined as the set of all families $(h_i)_{i \in I}$ with $h_i \in H_i$ such that $\sum_{i \in I} \|h_i\|_{H_i}^2 < \infty$, equipped with the inner product $\langle (h_i), (k_i) \rangle = \sum_{i \in I} \langle h_i, k_i \rangle_{H_i}$. The associated [norm](/page/Norm) is $\|(h_i)\| = \sqrt{\sum_{i \in I} \|h_i\|^2}$, and this space is complete, hence a [Hilbert space](/page/Hilbert_space), when the [index set](/page/Index_set) $I$ is countable (or finite). For finite sums, the condition reduces to all but finitely many $h_i = 0$, but the infinite case requires the square-summable [norm](/page/Norm) condition to ensure [convergence](/page/Convergence).[](https://spot.colorado.edu/~baggett/funcchap8.pdf)[](https://www.math.ucdavis.edu/~hunter/book/ch6.pdf) Each [Hilbert space](/page/Hilbert_space) $H_i$ embeds into the [direct sum](/page/Direct_sum) via the orthogonal inclusion $\iota_i: H_i \to \bigoplus_{j \in I} H_j$ defined by $\iota_i(h) = (0, \dots, h, \dots, 0)$ with $h$ in the $i$-th position, and these embeddings satisfy $\iota_i(H_i) \perp \iota_j(H_j)$ for $i \neq j$. This yields an orthogonal decomposition $\bigoplus_{i \in I} H_i = \overline{\sum_{i \in I} \iota_i(H_i)}$, where the closure is taken in the [direct sum](/page/Direct_sum) norm. An illustrative example is the finite [direct sum](/page/Direct_sum) $L^2(\mathbb{R}) \oplus L^2(\mathbb{R})$, which is isometrically isomorphic to $L^2(\mathbb{R} \times \{1,2\})$ under the [product measure](/page/Product_measure) (Lebesgue on $\mathbb{R}$ and [counting on](/page/Counting_On) $\{1,2\}$), via the map sending $(f,g)$ to the function that is $f$ on $\mathbb{R} \times \{1\}$ and $g$ on $\mathbb{R} \times \{2\}$. Moreover, the [direct sum](/page/Direct_sum) of countably many separable [Hilbert space](/page/Hilbert_space)s is separable.[](https://www.math.ucdavis.edu/~hunter/book/ch6.pdf) Bounded linear operators on the direct sum can be constructed componentwise: given bounded operators $T_i: H_i \to H_i$ with $\sup_{i \in I} \|T_i\| < \infty$, the direct sum operator $T = \bigoplus_{i \in I} T_i$ acts by $T((h_i)) = (T_i h_i)$ and is bounded on $\bigoplus H_i$ with $\|T\| = \sup_i \|T_i\|$. A key property extending [Parseval's identity](/page/Parseval's_identity) is that for any $(h_i) \in \bigoplus H_i$, \left| \sum_{i \in I} \iota_i(h_i) \right|^2 = \sum_{i \in I} |h_i|^2, which follows directly from the inner product definition and [orthogonality](/page/Orthogonality) of the $\iota_i(H_i)$. For [infinite](/page/Infinite) direct sums, the square-summable norm condition ensures that only families with $\sum \|h_i\|^2 < \infty$ are included, analogous to the $\ell^2$ direct sum over scalars.[](https://www.math.ucdavis.edu/~hunter/book/ch8.pdf)[](https://spot.colorado.edu/~baggett/funcchap8.pdf)

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