Definition:Internal Direct Sum of Rings/Direct Summand

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $S_1, S_2, \ldots, S_n$ be a sequence of subrings of $R$.

Let $\displaystyle S = \prod_{j \mathop = 1}^n S_j$ be the ring direct sum of $S_1$ to $S_n$.

In Conditions for Internal Ring Direct Sum it is proved that for this to be the case, then $S_1, S_2, \ldots, S_n$ must be ideals of $R$.

Such ideals are known as direct summands of $R$.