\begin{definition}[Definition:Internal Direct Sum 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 $S_1, S_2, \ldots, S_n$ {{iff}} 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 $R$.
\end{definition}