Lemma 11.4.5. Let $R$ be a possibly noncommutative ring. Let $n \geq 1$ be an integer. Let $R_ n = \text{Mat}(n \times n, R)$.

The functors $M \mapsto M^{\oplus n}$ and $N \mapsto Ne_{11}$ define quasi-inverse equivalences of categories $\text{Mod}_ R \leftrightarrow \text{Mod}_{R_ n}$.

A two-sided ideal of $R_ n$ is of the form $IR_ n$ for some two-sided ideal $I$ of $R$.

The center of $R_ n$ is equal to the center of $R$.

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