Semilocal ring

Remarks

By the Artin–Wedderburn theorem, $R$ is semilocal exactly when $R/J(R) \cong M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k)$ for division rings $D_i$ and integers $n_i \ge 1$. Example: $M_n(\Z_{(p)})$ is semilocal, since its Jacobson radical is $M_n(p\Z_{(p)})$ and the quotient is $M_n(\mathbb{F}_p)$. Non-example: $\Z$ is not semilocal: $J(\Z) = 0$ and $\Z$ is not semisimple, since the ideal $2\Z$ has no complement. A commutative ring is semilocal if and only if it has finitely many maximal ideals; this is the commutative notion. The definition is left-right symmetric, and the spelling semi-local is also used.