direct sum

《代数学引论》 · 第 83 页

我们称这个群为群 $G_1$ 与 $G_2$ 的直和，记为 $G_1\oplus G_2$。

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《Algebra》 · 第 173 页 · Chapter IV，Section 1「Modules, Homomorphisms and Exact Sequences」，Theorem 1.11 及其后定名段

Theorem 1.11. Let $R$ be a ring and $\{A_i\mid i\in I\}$ a nonempty family of $R$-modules, $\prod_{i\in I}A_i$ the direct product of the abelian groups $A_i$, and $\sum_{i\in I}A_i$ the direct sum of the abelian groups $A_i$. (i) $\prod_{i\in I}A_i$ is an $R$-module with the action of $R$ given by $r\{a_i\}=\{ra_i\}$. (ii) $\sum_{i\in I}A_i$ is a submodule of $\prod_{i\in I}A_i$. (iii) For each $k\in I$, the canonical projection $\pi_k:\prod A_i\to A_k$ (Theorem I.8.1) is an $R$-module epimorphism. (iv) For each $k\in I$, the canonical injection $\iota_k:A_k\to\sum A_i$ (Theorem I.8.4) is an $R$-module monomorphism. […] $\prod_{i\in I}A_i$ is called the (external) direct product of the family of $R$-modules $\{A_i\mid i\in I\}$ and $\sum_{i\in I}A_i$ is its (external) direct sum. If the index set is finite, say $I=\{1,2,\ldots,n\}$, then the direct product and direct sum coincide and will be written $A_1\oplus A_2\oplus\cdots\oplus A_n$.