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 finite sequence of subrings of $R$.

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

Then $S$ is the 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 $R$.

Also known as

 * The ring direct sum is also known as the internal ring direct sum or internal direct sum.

Also see

 * Definition:Internal Direct Product


 * Definition:External Direct Product
 * Definition:External Direct Sum of Rings