Semi-local ring
= Semi-local ring
{title2=$R/J(R)\text{ is semisimple Artinian}$}
{wiki}
= Semilocal ring
{synonym}
A <ring> is semilocal when its quotient by the <Jacobson radical> is a <semisimple ring> that is <Artinian>. Equivalently, that quotient is a <right Artinian ring>: it has zero <Jacobson radical>, and the Artinian structure theorem makes it semisimple. A commutative <ring> is semilocal exactly when it has finitely many <maximal ideals>.