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 $S_1, S_2, \ldots, S_n$ 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 internal direct sum (of rings) is also known as the ring direct sum or the internal ring direct sum.

Some sources give this as internal direct product (of rings).

Also see

 * Definition:Internal Direct Product


 * Definition:External Direct Product
 * Definition:Ring Direct Product
 * Definition:Ring Direct Sum