# 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 $S_1, S_2, \ldots, S_n$ if and only if 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 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 $R$.

## 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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 22$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 3.1$: Direct sums: Definition $3.2$