# Endomorphism ring
An **endomorphism ring** is the ring formed by all endomorphisms of a mathematical object, that is, all homomorphisms from the object to itself, with addition of maps as the addition and composition of maps as the multiplication. For an object \( A \) in an [additive category](https://www.edgepedia.org/additive-category), the ring is written \( \mathop{\rm End} A = \mathop{\rm Hom}(A, A) \); the identity morphism \( 1_A \) serves as the unit element.[1](https://encyclopediaofmath.org/wiki/Endomorphism_ring) The most common case is the endomorphism ring \( \mathop{\rm End}_R(M) \) of a module \( M \) over a ring \( R \), which collects the \( R \)-linear maps \( M \to M \).[2](https://mathworld.wolfram.com/EndomorphismRing.html)
| Key fact | Detail |
|---|---|
| Definition | \( \mathop{\rm End} A = \mathop{\rm Hom}(A, A) \) for an object \( A \) of an additive category; multiplication is composition of morphisms, addition is the addition defined by the additive category axioms, and \( 1_A \) is the unit.[1](https://encyclopediaofmath.org/wiki/Endomorphism_ring) |
| Units | An element \( \phi \in \mathop{\rm End} A \) is invertible if and only if \( \phi \) is an automorphism of \( A \).[1](https://encyclopediaofmath.org/wiki/Endomorphism_ring) |
| Module case | \( \mathop{\rm End}_R(M) \) is generally noncommutative but always a ring with unit, the identity map on \( M \); it is a subring of \( \mathop{\rm End}_{\mathbb Z}(M) \), the endomorphism ring of the underlying abelian group.[2](https://mathworld.wolfram.com/EndomorphismRing.html) • [3](https://web.mit.edu/18.705/www/13Ed-2up.pdf) |
| Ring as its own endomorphisms | Left multiplications identify \( R \cong \mathop{\rm End}(R_R) \), the endomorphism ring of \( R \) as a right module over itself.[4](https://www.kevos.com/engineering/matrix-rings-and-endomorphism-rings) • [3](https://web.mit.edu/18.705/www/13Ed-2up.pdf) |
| Simple modules | By Schur's lemma, \( \mathop{\rm End}_R(M) \) of a simple module \( M \) is a division ring, and \( \mathop{\rm End}_R(M^n) \cong M_n(D) \) where \( D = \mathop{\rm End}_R(M) \).[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html) |
| Morita theory | Two algebras \( A \) and \( B \) are Morita equivalent when there is a progenerator in \( \mathop{\rm mod} A \) with endomorphism ring \( B \).[6](https://www.math.uni-bielefeld.de/~ringel/lectures/padua/morita.htm) |
| Center | The center \( C(R) \) of a unital associative ring \( R \) is isomorphic to the endomorphism ring of \( R \) viewed as a bimodule over itself, that is, a left module over \( R \otimes R^{\rm op} \).[7](https://arxiv.org/abs/1207.4228) |
## Definition and basic structure
The ring structure is defined in two operations. The ring operations on \( \mathop{\rm End} A \) come from the additive category's axioms: composing morphisms serves as multiplication, while their sum, also prescribed by those axioms, serves as addition.[1](https://encyclopediaofmath.org/wiki/Endomorphism_ring) For a module \( M \) over a unit ring \( R \), the addition is pointwise, \( (f+g)(x) = f(x) + g(x) \), and the product is composition, \( (fg)(x) = f(g(x)) \).[2](https://mathworld.wolfram.com/EndomorphismRing.html) The resulting ring is in general noncommutative, but it always has a unit element, namely the identity map on \( M \).[2](https://mathworld.wolfram.com/EndomorphismRing.html)
Invertibility inside the ring is exactly automorphism: an element \( \phi \) of \( \mathop{\rm End} A \) is invertible if and only if \( \phi \) is an automorphism of the object \( A \).[1](https://encyclopediaofmath.org/wiki/Endomorphism_ring) The group of units of the endomorphism ring is therefore the automorphism group of the object.
**Left and right operator rings.** When homomorphisms of an abelian group \( M \) are written as operations from the left, the product of \( f, g : M \to M \) is \( f \circ g(m) = f(g(m)) \); writing on the right instead gives \( (m)g * f = ((m)g)f \). The endomorphism ring acting from the left is denoted \( \mathop{\rm End}_l(M) \) and from the right \( \mathop{\rm End}_r(M) \), and there is an anti-isomorphism \( \mathop{\rm End}_l(M) \to \mathop{\rm End}_r(M) \) sending \( f \circ g \) to \( g * f \). In practice one writes \( \mathop{\rm End}(M) \) for either and lets context decide.[8](https://library.uc.edu.kh/userfiles/pdf/40.Foundations%20of%20module%20and%20ring%20theory%20a%20handbook%20for%20study%20and%20research.pdf)
## Endomorphism rings of modules
When A is an R-module, End A is made up of precisely those endomorphisms of the underlying abelian group that are compatible with the action of every element of R.[1](https://encyclopediaofmath.org/wiki/Endomorphism_ring) Equivalently, \( \mathop{\rm End}_R(M) \) is a subring of \( \mathop{\rm End}_{\mathbb Z}(M) \), the endomorphism ring of \( M \) as a bare abelian group.[3](https://web.mit.edu/18.705/www/13Ed-2up.pdf) Stronger module structure leaves fewer endomorphisms, so the ring shrinks as \( R \) grows.
The ring \( R \) itself acts on any module by multiplication. For \( x \in R \), the map \( \mu_x : M \to M \) defined by \( \mu_x(m) = xm \) is an endomorphism, and \( x \mapsto \mu_x \) is a ring map \( \mu_R : R \to \mathop{\rm End}_R(M) \subset \mathop{\rm End}_{\mathbb Z}(M) \), whose kernel is the annihilator \( \mathop{\rm Ann}(M) \).[3](https://web.mit.edu/18.705/www/13Ed-2up.pdf) Conversely, given an abelian group \( N \) and a ring map \( \nu : R \to \mathop{\rm End}_{\mathbb Z}(N) \), setting \( xn = (\nu x)(n) \) makes \( N \) an \( R \)-module.[3](https://web.mit.edu/18.705/www/13Ed-2up.pdf) A module structure on an abelian group and a ring homomorphism into an endomorphism ring are the same data.
**Every ring is an endomorphism ring.** Taking \( M = R \) as a right module over itself, left multiplications identify \( R \cong \mathop{\rm End}(R_R) \), and every matrix ring is the endomorphism ring of a free module.[4](https://www.kevos.com/engineering/matrix-rings-and-endomorphism-rings) This is why endomorphism rings can serve as a concrete model for the abstract theory of rings.
## Functoriality
Endomorphism rings behave naturally under maps of categories. Given two objects \( A \) and \( B \) in an additive category \( C \), the abelian group \( \mathop{\rm Hom}(A, B) \) is naturally a right \( \mathop{\rm End} A \)-module and a left \( \mathop{\rm End} B \)-module. A covariant additive functor \( T : C \to C_1 \) induces a natural homomorphism \( \mathop{\rm End} A \to \mathop{\rm End} T(A) \) for each object \( A \), and a contravariant functor induces an anti-homomorphism.[1](https://encyclopediaofmath.org/wiki/Endomorphism_ring)
## Simple modules and Schur's lemma
The most quoted structural result about endomorphism rings is [Schur's lemma](https://www.edgepedia.org/schurs-lemma): if \( M \) and \( N \) are simple \( R \)-modules, then any nonzero homomorphism \( f : M \to N \) is an isomorphism, because its kernel is a proper submodule of \( M \) and its image is a nonzero submodule of \( N \).[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html) Applied to endomorphisms, every nonzero \( R \)-linear endomorphism of a simple module is invertible, so \( \mathop{\rm End}_R(M) \) is a division ring.[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)
The division ring need not be commutative. Over a commutative ring, the endomorphism ring of a module of rank one is commutative, but over a general ring the endomorphism ring of a simple module can be noncommutative; \( \mathop{\rm End}_{\mathbb H}(\mathbb H) \cong \mathbb H \), the quaternions, is the standard example.[4](https://www.kevos.com/engineering/matrix-rings-and-endomorphism-rings)
**Direct sums.** For a simple module \( M \) and \( n \ge 1 \), \( \mathop{\rm End}_R(M^n) \cong M_n(D) \), where \( D = \mathop{\rm End}_R(M) \); the isomorphism records an endomorphism of \( M^n \) as a matrix of maps \( \varphi_{ij} = \pi_j \circ \varphi \circ \iota_i \) built from projections and inclusions.[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html) More generally, if \( M \cong N_1^{n_1} \oplus \cdots \oplus N_k^{n_k} \) with the \( N_i \) mutually nonisomorphic simple modules, then \( \mathop{\rm End}_R(M) \cong \prod_{i=1}^{k} M_{n_i}(D_i) \), since \( \mathop{\rm Hom}_R(N_i, N_j) = 0 \) for \( i \neq j \).[5](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html) These identifications, together with \( R \cong \mathop{\rm End}(R_R) \) and \( \mathop{\rm End}_R(M^n) \cong M_n(\mathop{\rm End}_R(M)) \), are the foundation on which Wedderburn-Artin theory is built.[4](https://www.kevos.com/engineering/matrix-rings-and-endomorphism-rings)
## Role in module and representation theory
[Nathan Jacobson](https://www.edgepedia.org/nathan-jacobson) gave special attention in his 1943 monograph *The Theory of Rings* to rings of endomorphisms: through the regular representations, abstract ring theory emerges as a special case of the more concrete theory of endomorphisms, and the theory of modules—and therefore representation theory—can be viewed as the study of a set of endomorphism rings, each a homomorphic image of a fixed ring \( R \).[9](https://mathshistory.st-andrews.ac.uk/Extras/Jacobson_rings/) The representation-theoretic line itself goes back to [Emmy Noether](https://www.edgepedia.org/emmy-noether), who reformulated the original problem of representing a group by matrices as a theory of modules.[9](https://mathshistory.st-andrews.ac.uk/Extras/Jacobson_rings/)
Endomorphism rings also appear in ring-theoretic situations beyond module categories: as the center of a ring, as the fix ring of a ring with a group action, and as the subring of constants of a derivation.[7](https://arxiv.org/abs/1207.4228)
**Morita equivalence.** Two algebras \( A \) and \( B \) are called Morita equivalent when the categories \( \mathop{\rm mod} A \) and \( \mathop{\rm mod} B \) are equivalent, which holds when there exists a progenerator \( P \) in \( \mathop{\rm mod} A \), a module that is finitely generated, projective, and a generator, with endomorphism ring \( B \); the functor \( \mathop{\rm Hom}_A(P, -) \) then gives the equivalence.[6](https://www.math.uni-bielefeld.de/~ringel/lectures/padua/morita.htm) In the same spirit, for a fixed ring \( R \), all rings \( S \) arising as endomorphism rings of progenerators of \( R \) are exactly the rings Morita equivalent to \( R \).[10](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/lam2.pdf)
## Endomorphism rings in arithmetic geometry
Endomorphism rings of algebraic varieties carry arithmetic information. For an elliptic curve \( E \) without complex multiplication defined over a number field \( K \), and for all but finitely many nonarchimedean places \( v \) of \( K \), the reduction \( E(v) \) at \( v \) has an endomorphism ring \( \mathop{\rm End}(E(v)) \) that, when the reduction is ordinary, is an order in an imaginary quadratic field; the set of places with ordinary reduction has density 1, a result due to Serre.[11](https://arxiv.org/abs/1509.07095) Computing such rings, for example for supersingular elliptic curves over \( \mathbb F_{p^2} \), is an active algorithmic problem in its own right.[12](https://www.sciencedirect.com/science/article/pii/S0021869325000353)
## References
1. [Endomorphism ring, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Endomorphism_ring)
2. [Endomorphism Ring, Wolfram MathWorld](https://mathworld.wolfram.com/EndomorphismRing.html)
3. [MIT 18.705 lecture notes, section 4.4 (Endomorphisms)](https://web.mit.edu/18.705/www/13Ed-2up.pdf)
4. [Matrix Rings and Endomorphism Rings of Modules, KEVOS](https://www.kevos.com/engineering/matrix-rings-and-endomorphism-rings)
5. [Romyar Sharifi, Abstract Algebra, Chapter 13: Representation theory](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)
6. [Claus Michael Ringel, Padua lectures: Morita equivalence](https://www.math.uni-bielefeld.de/~ringel/lectures/padua/morita.htm)
7. [Endomorphism rings of modules over prime rings, arXiv:1207.4228](https://arxiv.org/abs/1207.4228)
8. [Foundations of Module and Ring Theory (Wisbauer), Chapter 2](https://library.uc.edu.kh/userfiles/pdf/40.Foundations%20of%20module%20and%20ring%20theory%20a%20handbook%20for%20study%20and%20research.pdf)
9. [Nathan Jacobson, The Theory of Rings (1943), preface via MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Jacobson_rings/)
10. [T. Y. Lam, Exercises in Modules and Rings](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/lam2.pdf)
11. [Endomorphism rings of reductions of elliptic curves and abelian varieties, arXiv:1509.07095](https://arxiv.org/abs/1509.07095)
12. [Computing supersingular endomorphism rings using inseparable endomorphisms, Journal of Algebra 668 (2025)](https://www.sciencedirect.com/science/article/pii/S0021869325000353)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Module theory › Tensor products and bimodules › Module homomorphisms*
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