Remark

Let A be a ring. Let $(a_i)_{i \in I}$ be a finite family of subgroups of the additive group $A^+$ of A such that $A^+$ is the direct sum of the $a_i$. Suppose $a_i a_i \subset ai$ for $i \in I$ and $a_i a_j = \{0\}$ for $i \neq j$. Then $a_i$ is for all $i \in I$ a two-sided ideal of A. With addition and multiplication induced by those on A, $a_i$ is a ring with unit element the component of $1 \in A$ in $a_i$. If $b_i = \sum_{j \neq i} a_j$, clearly the $b_i$ satisfy condition (c) of Proposition 10 and hence $(b_i)_{i \in I}$ is a direct decomposition of A, which is said to be defined by $(a_i)_{i \in I}$.