Endomorphism Ring - Properties

Properties

• Endomorphism rings always have multiplicative identity, namely the identity map.
• Endomorphism rings are typically non-commutative.
• If a module is simple, then its endomorphism ring is a division ring (this is sometimes called Schur's lemma).
• A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotents. If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring.
• 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 a an Artinian uniform module is a local ring.
• 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.
• 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.
• The formation of endomorphism rings can be viewed as a functor from the category of abelian groups (Ab) to the category of rings.