Endomorphism ring

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.

• 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).[4]
• A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotent elements.[5] If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring.[6]
• 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.[7]
• 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.[8]
• 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.

• 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 ${\displaystyle A}$, ${\displaystyle \mathrm {M} _{n}(\operatorname {End} (A))\cong \operatorname {End} (A^{n})}$, since any matrix in ${\displaystyle \mathrm {M} _{n}(\operatorname {End} (A))}$ carries a natural homomorphism structure of ${\displaystyle A^{n}}$ as follows:

  ${\displaystyle {\begin{pmatrix}\varphi _{11}&\cdots &\varphi _{1n}\\\vdots &&\vdots \\\varphi _{n1}&\cdots &\varphi _{nn}\end{pmatrix}}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{i=1}^{n}\varphi _{1i}(a_{i})\\\vdots \\\sum _{i=1}^{n}\varphi _{ni}(a_{i})\end{pmatrix}}.}$

One can use this isomorphism to construct many non-commutative endomorphism rings. For example: ${\displaystyle \operatorname {End} (\mathbb {Z} \times \mathbb {Z} )\cong \mathrm {M} _{2}(\mathbb {Z} )}$, since ${\displaystyle \operatorname {End} (\mathbb {Z} )\cong \mathbb {Z} }$.

Also, when ${\displaystyle R=K}$ is a field, there is a canonical isomorphism ${\displaystyle \operatorname {End} (K)\cong K}$, so ${\displaystyle \operatorname {End} (K^{n})\cong \mathrm {M} _{n}(K)}$, 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(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.

• Camillo, V. P.; Khurana, D.; Lam, T. Y.; Nicholson, W. K.; Zhou, Y. (2006), "Continuous modules are clean", J. Algebra, 304 (1): 94–111, doi:10.1016/j.jalgebra.2006.06.032, ISSN 0021-8693, MR 2255822
• Drozd, Yu. A.; Kirichenko, V.V. (1994), Finite Dimensional Algebras, Berlin: Springer-Verlag, ISBN 3-540-53380-X
• Dummit, David; Foote, Richard, Algebra
• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
• "Endomorphism ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Jacobson, Nathan (2009), Basic algebra, vol. 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7
• Passman, Donald S. (1991), A Course in Ring Theory, Pacific Grove: Wadsworth & Brooks/Cole, ISBN 0-534-13776-8
• Wisbauer, Robert (1991), Foundations of module and ring theory, Algebra, Logic and Applications, vol. 3 (Revised and translated from the 1988 German ed.), Philadelphia, PA: Gordon and Breach Science Publishers, pp. xii+606, ISBN 2-88124-805-5, MR 1144522 A handbook for study and research