Endomorphism ring

In abstract algebra, the endomorphism ring of an abelian group X, denoted by End(X), is the set of all homomorphisms of X into itself. The addition operation is defined by pointwise addition of functions and the multiplication operation is defined by function composition.

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 resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra.

Description

Let (A, +) be an abelian group and we consider the group homomorphisms from A into A. Then addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism (f + g)(x) := f(x) + g(x). Under this operation End(A) is an abelian group. With the additional operation of composition of homomorphisms, End(A) is a ring with multiplicative identity. This composition is explicitly (fg)(x) := f(g(x)). The multiplicative identity is the identity homomorphism on A.

If the set A does not form an abelian group, then the above construction is not necessarily additive, as then the sum of two homomorphisms need not be a homomorphism. This set of endomorphisms is a canonical example of a near-ring which is not a ring.

Properties

• Endomorphism rings always have additive and multiplicative identities, respectively the zero map and identity map.
• Endomorphism rings are associative, but 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 idempotent elements. 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 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.

Examples

• In the category of R modules the endomorphism ring of an R-module M will only use the R module homomorphisms, which are typically a proper subset of the abelian group homomorphisms. When M is a finitely generated projective module, the endomorphism ring is central to Morita equivalence of module categories.
• $End(\mathbb{Z}_2\times \mathbb{Z}_2, +)\cong M_2(\mathbb{Z}_2)$. The endomorphism ring of the additive abelian group $(\mathbb{Z}_2\times \mathbb{Z}_2, +)$ is isomorphic to the $2\times 2$ matrix ring over $\mathbb{Z}_2$. (see Dummit-Foote, Abstract Algebra 3rd edition, example (5), pp. 338 and example (5), pp. 346)
• If K is a field and we consider the K-vector space Kⁿ, then the endomorphism ring of Kⁿ consists of all K-linear maps from Kⁿ to Kⁿ: it is a K-algebra. After a basis for the vector space is chosen, this ring is naturally identified with the ring of n-by-n matrices with entries in K. More generally, the endomorphism algebra of the free module M = Rⁿ is naturally n-by-n matrices with entries in the ring R.
• As a particular example of the last point, for any ring R with unity, End(R_R) = R, where the elements of R act on R by left multiplication.
• In general, endomorphism rings can be defined for the objects of any preadditive category.

References

• A handbook for study and research