Category:Direct Sums of Rings

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This category contains results about Direct Sums of Rings.

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 $R$.

Pages in category "Direct Sums of Rings"

The following 2 pages are in this category, out of 2 total.