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 homomorphisms of X into itself.[1][2] 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[edit]

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 which is not a ring.

Properties[edit]

Examples[edit]

  • 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.
  • 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.[10] 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(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[edit]

  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. ^ Camillo et al. Zhou.
  9. ^ Abelian groups may also be viewed as modules over the ring of integers.
  10. ^ Drozd & Kirichenko 1994, pp. 23–31.

References[edit]

  • Drozd, Yu. A.; Kirichenko, V.V. (1994), Finite Dimensional Algebras, Berlin: Springer-Verlag, ISBN 3-540-53380-X 
  • Dummit, David; Foote, Richard, Algebra 
  • Wisbauer, Robert (1991), Foundations of module and ring theory, Algebra, Logic and Applications 3 (Revised and translated from the 1988 German edition ed.), Philadelphia, PA: Gordon and Breach Science Publishers, pp. xii+606, ISBN 2-88124-805-5, MR 1144522  A handbook for study and research