Editing Endomorphism ring (section)

== Properties ==
* Endomorphism rings always have additive and multiplicative [[identity element|identities]], respectively the [[zero map]] and [[identity function|identity map]].
* Endomorphism rings are [[associative]], but typically [[non-commutative ring|non-commutative]].
* If a module is [[simple module|simple]], then its endomorphism ring is a [[division ring]] (this is sometimes called [[Schur's lemma]]).{{sfn|ps=none|Jacobson|2009|loc=p. 118}}
* A module is [[indecomposable module|indecomposable]] if and only if its endomorphism ring does not contain any non-trivial [[idempotent element (ring theory)|idempotent element]]s.{{sfn|ps=none|Jacobson|2009|loc=p. 111, Prop. 3.1}} If the module is an [[injective module]], then indecomposability is equivalent to the endomorphism ring being a [[local ring]].{{sfn|ps=none|Wisbauer|1991|loc=p. 163}}
* For a [[semisimple module]], the endomorphism ring is a [[von Neumann regular ring]].
* The endomorphism ring of a nonzero right [[uniserial module]] has either one or two maximal right ideals. If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.
* The endomorphism ring of an Artinian [[uniform module]] is a local ring.{{sfn|ps=none|Wisbauer|1991|loc=p. 263}}
* The endomorphism ring of a module with finite [[composition length]] is a [[semiprimary ring]].
* The endomorphism ring of a [[continuous module]] or [[discrete module]] is a [[clean ring]].{{sfn|ps=none|Camillo|Khurana|Lam|Nicholson|2006}}
* If an ''R'' module is finitely generated and projective (that is, a [[progenerator]]), then the endomorphism ring of the module and ''R'' share all Morita invariant properties. A fundamental result of Morita theory is that all rings equivalent to ''R'' arise as endomorphism rings of progenerators.