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 $\displaystyle S = \prod_{j=1}^n S_j$ 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

 * External Direct Product
 * Internal Direct Product