The Stacks project

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

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

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

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

Proof. Part (1) proves itself. If $J \subset R_ n$ is a two-sided ideal, then $J = \bigoplus e_{ii}Je_{jj}$ and all of the summands $e_{ii}Je_{jj}$ are equal to each other and are a two-sided ideal $I$ of $R$. This proves (2). Part (3) is clear. $\square$

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