Definition:Internal Direct Sum of Rings

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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