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.
For a finitely generated module on the right over a semilocal ring, the displayed condition makes the induced module endomorphism on injective. This quotient is an Artinian module, so Artinian modules are co-Hopfian makes that map surjective. The remaining quotient module equals its product with , and Nakayama lemma makes it zero.
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A **semi-local ring** is a concept in commutative algebra that generalizes some ideas of local rings. A ring \( R \) is called a semi-local ring if it has a finite number of maximal ideals. This means that the set of maximal ideals of \( R \) is not necessarily just one (as in the case of local rings), but consists of a finite collection of such ideals.