Definition 1.3.1 (Product).label Let $R$ be a ring and $\seqi{A}$ be $R$-modules, then there exists $(A, \bracsn{\pi_i}_{i \in I})$ such that:

1. (1)

   For each $i \in I$, $\pi_{i} \in \hom(A; A_{i})$.

2. (U)

   For any $(B, \seqi{T})$ satisfying (1), there exists a unique $T \in \hom(B; A)$ such that the following diagram commutes:

   \[\xymatrix{ B \ar@{->}[rd]_{T_i} \ar@{->}[r]^{T} & A \ar@{->}[d]^{\pi_i} \\ & A_i }\]

   
   

The module $A = \prod_{i \in I}A_{i}$ is the product of $\seqi{A}$.