Editing Direct sum of modules (section)

== Properties ==

* The direct sum is a [[submodule]] of the [[direct product]] of the modules ''M''<sub>''i''</sub> {{harv|Bourbaki|1989|loc=§II.1.7}}.  The direct product is the set of all functions ''α'' from ''I'' to the disjoint union of the modules ''M''<sub>''i''</sub> with ''α''(''i'')∈''M''<sub>''i''</sub>, but not necessarily vanishing for all but finitely many ''i''. If the index set ''I'' is finite, then the direct sum and the direct product are equal.
* Each of the modules ''M''<sub>''i''</sub> may be identified with the submodule of the direct sum consisting of those functions which vanish on all indices different from ''i''. With these identifications, every element ''x'' of the direct sum can be written in one and only one way as a sum of finitely many elements from the modules ''M''<sub>''i''</sub>.
* If the ''M''<sub>''i''</sub> are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of the ''M''<sub>''i''</sub>. The same is true for the [[rank of an abelian group|rank of abelian groups]] and the [[length of a module|length of modules]].
* Every vector space over the field ''K'' is isomorphic to a direct sum of sufficiently many copies of ''K'', so in a sense only these direct sums have to be considered. This is not true for modules over arbitrary rings.
* The [[tensor product]] distributes over direct sums in the following sense: if ''N'' is some right ''R''-module, then the direct sum of the tensor products of ''N'' with ''M''<sub>''i''</sub> (which are abelian groups) is naturally isomorphic to the tensor product of ''N'' with the direct sum of the ''M''<sub>''i''</sub>.
* Direct sums are [[commutative]] and [[associative]] (up to isomorphism), meaning that it doesn't matter in which order one forms the direct sum.
* The abelian group of ''R''-[[Linear map|linear homomorphisms]] from the direct sum to some left ''R''-module ''L'' is naturally isomorphic to the [[direct product]] of the abelian groups of ''R''-linear homomorphisms from ''M''<sub>''i''</sub> to ''L'': <math display="block">\operatorname{Hom}_R\biggl( \bigoplus_{i \in I} M_i,L\biggr) \cong \prod_{i \in I}\operatorname{Hom}_R\left(M_i,L\right).</math> Indeed, there is clearly a [[homomorphism]] ''τ'' from the left hand side to the right hand side, where ''τ''(''θ'')(''i'') is the ''R''-linear homomorphism sending ''x''∈''M''<sub>''i''</sub> to ''θ''(''x'') (using the natural inclusion of ''M''<sub>''i''</sub> into the direct sum). The inverse of the homomorphism ''τ'' is defined by <math display="block"> \tau^{-1}(\beta)(\alpha) = \sum_{i\in I} \beta(i)(\alpha(i))</math> for any ''α'' in the direct sum of the modules ''M''<sub>''i''</sub>. The key point is that the definition of ''τ''<sup>−1</sup> makes sense because ''α''(''i'') is zero for all but finitely many ''i'', and so the sum is finite.{{pb}}In particular, the [[dual space|dual vector space]] of a direct sum of vector spaces is isomorphic to the [[direct product]] of the duals of those spaces.
*The ''finite'' direct sum of modules is a [[biproduct]]: If <math display="block">p_k: A_1 \oplus \cdots \oplus A_n \to A_k</math> are the canonical projection mappings and <math display="block">i_k: A_k \mapsto A_1 \oplus \cdots \oplus A_n </math> are the inclusion mappings, then <math display="block">i_1 \circ p_1 + \cdots + i_n \circ p_n</math> equals the identity morphism of ''A''<sub>1</sub> ⊕ ⋯ ⊕ ''A''<sub>''n''</sub>, and <math display="block">p_k \circ i_l</math> is the identity morphism of ''A''<sub>''k''</sub> in the case ''l'' = ''k'', and is the zero map otherwise.