Endomorphism ring

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In abstract algebra, the endomorphism ring of an abelian group X, denoted by End(X), is the set of all endomorphisms of X (i.e., the set of all homomorphisms of X into itself) endowed with an addition operation defined by pointwise addition of functions and a multiplication operation defined by function 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 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.[3] This set of endomorphisms is a canonical example of a near-ring that is not a ring.

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.[9] When M is a finitely generated projective module, the endomorphism ring is central to Morita equivalence of module categories.
• For any abelian group ${\displaystyle A}$, ${\displaystyle M_{n}(End(A))\cong End(A^{n})}$, since any matrix in ${\displaystyle M_{n}(End(A))}$ carries a natural homomorphism structure of ${\displaystyle A^{n}}$ as follows:
${\displaystyle {\begin{pmatrix}\phi _{11}&\cdots &\phi _{1n}\\\vdots &&\vdots \\\phi _{n1}&\cdots &\phi _{nn}\end{pmatrix}}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{i=1}^{n}\phi _{1i}(a_{i})\\\vdots \\\sum _{i=1}^{n}\phi _{ni}(a_{i})\end{pmatrix}}}$.
One can use this isomorphism to construct a lot of non-commutative endomorphism rings. For example: ${\displaystyle End(\mathbb {Z} \times \mathbb {Z} )\cong M_{2}(\mathbb {Z} )}$, since ${\displaystyle End(\mathbb {Z} )\cong \mathbb {Z} }$.
Also, when ${\displaystyle R=K}$ is a field, there is a canonical isomorphism ${\displaystyle End(K)\cong K}$, so ${\displaystyle End(K^{n})\cong M_{n}(K)}$, that is, the endomorphism ring of a ${\displaystyle K}$-vector space is identified with the ring of n-by-n matrices with entries in ${\displaystyle K}$.[10] More generally, the endomorphism algebra of the free module ${\displaystyle M=R^{n}}$ is naturally ${\displaystyle n}$-by-${\displaystyle n}$ matrices with entries in the ring ${\displaystyle R}$.
• As a particular example of the last point, for any ring R with unity, End(RR) = 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.

Notes

1. ^ Fraleigh (1976, p. 211)
2. ^ Passman (1991, pp. 4–5)
3. ^ Dummit & Foote, p. 347)
4. ^ Jacobson 2009, p. 118.
5. ^ Jacobson 2009, p. 111, Prop. 3.1.
6. ^ Wisbauer 1991, p.163.
7. ^ Wisbauer 1991, p. 263.
8. ^
9. ^ Abelian groups may also be viewed as modules over the ring of integers.
10. ^ Drozd & Kirichenko 1994, pp. 23–31.