In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all...

In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For...

with ring homomorphisms as morphisms (see Category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism...

R (assumed unital), the endomorphism ring EndR(R) = R, where each endomorphism arises as left multiplication by a fixed ring element. With this modification...

the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms. More...

multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory...

states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent...

field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p...

multiplication given by function composition, called the endomorphism ring of M. The group of units of this ring is the automorphism group of M. Schur's lemma says...

properties of the endomorphism ring of M. A module is said to be strongly indecomposable if its endomorphism ring is a local ring. For the important...

mathematics. More generally, endomorphism rings of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the Klein four-group...

elements except identity. Endomorphism A group homomorphism, h: G → G; the domain and codomain are the same. Also called an endomorphism of G. Automorphism A...

Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings. Specifically, if...

homomorphism is a semiprimitive ring, and every semiprimitive ring is isomorphic to such an image. The endomorphism ring of a semisimple module is not only...

zero ring. The direct product of an empty collection of rings is the zero ring. The endomorphism ring of the trivial group is the zero ring. The ring of...

multiplication by x behaves normally except that x·1 = 0. The endomorphism ring is simply the ring of formal power series. If G is a finite group and k a field...

{K}}} ) has rank 4, while the endomorphism group of every other elliptic curve has only rank 1 or 2. The endomorphism ring of a supersingular elliptic curve...

commutative domain is called an integral domain. endomorphism An endomorphism ring is a ring formed by the endomorphisms of an object with additive structure; the...

(Lam 2001, §1,ex1.9). The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from...

algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M is either an automorphism or nilpotent...

the endomorphism ring of the additive group of Z is isomorphic to the ring Z. Its automorphism group is isomorphic to the group of units of the ring Z,...

division ring. In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring; every...

known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector...

indeed a semisimple ring. The classical ring of quotients for any commutative Noetherian ring is a semilocal ring. The endomorphism ring of an Artinian module...

of H. The endomorphism ring of an indecomposable injective module is local and thus Azumaya's theorem says that, over a left Noetherian ring, each indecomposable...

and 1. The endomorphism ring of a continuous module is a clean ring. Every clean ring is an exchange ring. A matrix ring over a clean ring is itself clean...

that the endomorphism hom-set Hom(A,A) is a ring, if we define multiplication in the ring to be composition. This ring is the endomorphism ring of A. Conversely...

a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse...

homomorphism or an isomorphism. Consequently, the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma. The converse...

the endomorphism ring EndS(M) is von Neumann regular. In particular, every semisimple ring is von Neumann regular. Indeed, the semisimple rings are precisely...