Definition:Internal Direct Sum of Rings

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 $$S = \prod_{j=1}^n S_j$$ be the cartesian product of $$S_1$$ to $$S_n$$.

Then $$S$$ is the ring direct sum (or internal (ring) direct sum) of $$S_1, S_2, \ldots, S_n$$ iff the mapping $$\phi: S \to R$$ defined as:
 * $$\phi\left({\left({x_1, x_2, \ldots, x_n}\right)}\right) = x_1 + x_2 + \cdots x_n$$

is an isomorphism from $$S$$ to $$R$$.

Direct Summand
In Conditions for Internal 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 $$R$$.

Also see

 * Internal Direct Product