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Let $S$ be an arbitrary ring. We write $R = M_n(S)$ for the ring of $n\times n$ matrices with entries in $S$. There is an obvious identification of $S$ with the scalar matrices $S\cdot I_n$. We will also write $E_{ii}$ for the matrix unit with $1$ on the $(i,i)$-entry and zeros everywhere else. Note that it is very clear that

and also that

Proposition 17.4

[proposition] The subring $S \subseteq R = M_n(S)$ is the centralizer (in $R$) of the of the set of matrix units $\{E_{ij}\}$. [/proposition]

Definition (matrix units)

[definition] We say that a set

in a ring $R$ is a set of matrix units if

for all $i, j, k, \ell$. If we also have the condition that $\sum_{i=1}^n e_{ii} = 1_R$, then we say that $\{e_{ij}\}$ is a full set of matrix units in $R$. [/definition]

Theorem 17.5

[theorem] For any ring $R$ and any fixed integer $n \ge 1$, TFAE:

1. $R\cong M_n(S)$ for some ring $S$.
2. $R$ has a full set of matrix $\{e_{ij} \mid 1 \le i,j \le n\}$.
3. $R = \mathfrak{I}_1 \oplus \cdots \oplus \mathfrak{I}_n$ for suitable right ideals $\mathfrak{I}_i$ which are mutually isomorphisc as right $R$-modules. [/theorem]

As a remark, the ideals $\mathfrak{I}_i \coloneqq e_{i1}R$.

Corollary 17.7

[corollary] Let $f:R \to R’$ be a ring homomorphism, where $R = M_n(S)$ for some ring $S$. Then $R’$ can be expressed in the form $M_n(S’)$ for some ring $S’$ such that $f$ is “induced” by some ring homomorphism $f_0:S \to S’$.

In particular, any ring containing $M_n(S)$ has the form $M_n(S’)$ for some ring $S’ \supseteq S$. [/corollary]

Corollary 17.8

[corollary] Let $I$ be an ideal in $R = M_n(S)$. Then $I = M_n(\mathfrak{I})$ for some ideal $\mathfrak{I}$ in $S$. [/corollary]

Theorem 17.9

[theorem] Let $S = \End_A(P)$, where $P$ is a right $A$-module. Then $S \cong M_n(T)$ for some ring $T$ iff $P \cong nQ$ for some right $A$-module $Q$. [/theorem]