Definition 1.3.2 (Direct Sum).label Let $E$ be a ring and $\seqi{A}$ be $R$-modules, then there exists $(A, \bracsn{\iota_i}_{i \in I})$ such that:
- (1)
For each $i \in I$, $\iota_{i} \in \hom(A_{i}; A)$.
- (U)
For each $(B, \seqi{T})$ satisfying (1), there exists a unique $T \in \hom(A; B)$ such that the following diagram commutes
\[\xymatrix{ A \ar@{->}[r]^{T} & B \\ A_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & }\]
The module $A = \bigoplus_{i \in I}A_{i}$ is the direct sum of $\seqi{A}$.
Proof. Let
\[A = \bracs{x \in \prod_{i \in I}A_i \bigg | x_i \ne 0 \quad \text{for finitely many}\ i \in I}\]
For each $i \in I$, let
\[\iota_{i}: A_{i} \to A \quad (\iota_{i}x)_{j} = \begin{cases}x &i = j \\ 0 &i \ne j\end{cases}\]
then $\iota_{i} \in \hom(A_{i}; A)$.
(U): Let
\[T: A \to B \quad x \mapsto \sum_{i \in I}T_{i}x_{i}\]
then $T \in \hom(A; B)$ and the diagram commutes. Since $\bigcup_{i \in I}\iota_{i}(A_{i})$ spans $A$, $T$ is the unique linear map making the diagram commute.$\square$