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...

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...

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...

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

group ring of a finite group. However, even over the ring of integers, the module of rational numbers has an endomorphism ring that is a division ring, specifically...

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

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

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

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...

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

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...

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...

(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...

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...

n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn. If the ring R is commutative, that is...

and quaternions. SQIsign's security relies on the hardness of the endomorphism ring problem, which is currently considered hard. The authors also provide...

{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...

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 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...

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...

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 endomorphism ring is...

Antiautomorphism Automorphism (in Sudoku puzzles) Characteristic subgroup Endomorphism ring Frobenius automorphism Morphism Order automorphism (in order theory)...

Z-modules are equivalent. Any ring of characteristic n is a (Z/nZ)-algebra in the same way. Given an R-module M, the endomorphism ring of M, denoted EndR(M) is...

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the...

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...

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...

Noetherian (whereas every Artinian ring is Noetherian). The endomorphism ring of Z(p∞) is isomorphic to the ring of p-adic integers Zp. In the theory...