Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End(X). As the resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra.
[edit] Examples
The elements of the endomorphism ring of an abelian group (A, +) are the endomorphisms of A, i.e. the group homomorphisms from A to A. Any two such endomorphisms f and g can be added pointwise (using the formula (f+g)(x) = f(x) + g(x)), and the result f+g is again an endomorphism of A. Furthermore, f and g can also be composed to yield the endomorphism f o g, and this multiplication distributes over pointwise addition. The set of all endomorphisms of A, together with this addition and multiplication, satisfies all the axioms of a ring. This is the endomorphism ring of A. Its multiplicative identity is the identity map on A. Endomorphism rings are typically non-commutative.
The above construction does not work for groups that are not abelian: the sum of two homomorphisms need not be a homomorphism in that case.[1]
We can define the endomorphism ring of any module in exactly the same way, using module homomorphisms instead of abelian group homomorphisms; abelian groups are exactly modules over the integers. The result is an algebra over the ring R of scalar transformations.
If K is a field and we consider the K-vector space Kn, then the endomorphism ring of Kn (which consists of all K-linear maps from Kn to Kn) is naturally identified with the ring of n-by-n matrices with entries in K.[2] More generally, the endomorphism algebra of the free module M = Rn is naturally n-by-n matrices with entries in R.
In general, endomorphism rings can be defined for the objects of any preadditive category.
[edit] Properties
One can often translate properties of an object into properties of its endomorphism ring. For instance:
- If a module is simple, then its endomorphism ring is a division ring (this is sometimes called Schur's lemma).[3][4]
- A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotents.[5][6]
[edit] References
- ^ (Dummit Foote, p. 347)
- ^ (Drozd & Kirichenko 1994, pp. 23–24)
- ^ (Drozd & Kirichenko 1994, p. 31)
- ^ Jacobson (2009), p. 118.
- ^ (Drozd & Kirichenko 1994, p. 25)
- ^ Jacobson (2009), p. 111, Prop. 3.1.
- Drozd, Yu. A.; Kirichenko, V.V. (1994), Finite Dimensional Algebras, Berlin: Springer-Verlag, ISBN 354053380X
- Dummit, David; Foote, Richard, Algebra
- Jacobson, Nathan (2009), Basic algebra, 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7
