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 homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity.[1][2]
The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the category of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the endomorphism ring is often an algebra over some ring R, this may also be called the endomorphism algebra.
An abelian group is the same thing as a module over the ring of integers, which is the initial object in the category of rings. In a similar fashion, if R is any commutative ring, the endomorphisms of an R-module form an algebra over R by the same axioms and derivation. In particular, if R is a field, its modules M are vector spaces and the endomorphism ring of each is an algebra over the field R.
In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphismring 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 ringendomorphism, ring isomorphism, and ring automorphism...
field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p...
multiplication (CM) is the theory of elliptic curves E that have an endomorphismring larger than the integers. Put another way, it contains the theory...
mathematics. More generally, endomorphismrings of abelian groups are rarely commutative, the simplest example being the endomorphismring of the Klein four-group...
homomorphism is a semiprimitive ring, and every semiprimitive ring is isomorphic to such an image. The endomorphismring of a semisimple module is not only...
Non-commutative local rings arise naturally as endomorphismrings in the study of direct sum decompositions of modules over some other rings. Specifically, if...
zero ring. The direct product of an empty collection of rings is the zero ring. The endomorphismring of the trivial group is the zero ring. The ring of...
(Lam 2001, §1,ex1.9). The skew-polynomial ring is defined similarly for a ring R and a ringendomorphism f of R, by extending the multiplication from...
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...
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...
{K}}} ) has rank 4, while the endomorphism group of every other elliptic curve has only rank 1 or 2. The endomorphismring of a supersingular elliptic curve...
multiplication by x behaves normally except that x·1 = 0. The endomorphismring is simply the ring of formal power series. If G is a finite group and k a field...
known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphismrings of vector...
the endomorphismring 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 endomorphismring of S is a division ring; every...
commutative domain is called an integral domain. endomorphism An endomorphismring is a ring formed by the endomorphisms of an object with additive structure; the...
states that if a module has an decomposition into modules with local endomorphismrings, then all decompositions into indecomposable modules are equivalent...
function composition of endomorphisms over any commutative monoid. The theory of (associative) algebras over commutative rings can be generalized to one...
multiplication given by function composition, called the endomorphismring of M. The group of units of this ring is the automorphism group of M. Schur's lemma says...
of H. The endomorphismring of an indecomposable injective module is local and thus Azumaya's theorem says that, over a left Noetherian ring, each indecomposable...
homomorphism or an isomorphism. Consequently, the endomorphismring of any simple module is a division ring. This result is known as Schur's lemma. The converse...
unital associative algebra A has a natural homomorphism to its own endomorphismring End(A). A bilinear form can be defined on A in the sense of the previous...
the endomorphismring EndS(M) is von Neumann regular. In particular, every semisimple ring is von Neumann regular. Indeed, the semisimple rings are precisely...
idempotent endomorphism of M, then M is the direct sum of ker(f) and im(f).) A module of finite length is indecomposable if and only if its endomorphismring is...
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