Category:Direct Sums of Rings

This category contains results about Direct Sums of Rings.

Let $\struct {R, +, \circ}$ be a ring.

Let $S_1, S_2, \ldots, S_n$ be a finite sequence of subrings of $R$.

Let $\ds S = \prod_{j \mathop = 1}^n S_j$ 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:

$\map \phi {\left({x_1, x_2, \ldots, x_n} }\right) = x_1 + x_2 + \cdots x_n$

is an isomorphism from $S$ to $R$.