Definition:Internal Direct Sum of Rings/Direct Summand

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Let $\struct {R, +, \circ}$ be a ring.

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

Let $\ds S = \prod_{j \mathop = 1}^n S_j$ 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 $R$.