Let $R$ be a ring and $\mathfrak{a}_1$ , $\mathfrak{a}_2$ , ..., $\mathfrak{a}_n$ its ideals (left, right or two-sided). We say that $R$ is the internal direct sum of these ideals, denoted by $$R = \mathfrak{a}_1\oplus\mathfrak{a}_2\oplus\cdots\oplus\mathfrak{a}_n,$$ if both of the following conditions are true: $$R = \mathfrak{a}_1+\mathfrak{a}_2+\cdots+\mathfrak{a}_n,$$ $$\mathfrak{a}_i\cap\sum_{j\neq i}\mathfrak{a}_j = \{0\}\quad\forall i.$$