Endomorphism ring

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In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object.

We will start with the example of abelian groups. Suppose A is an abelian group. As the name suggests, the elements of the endomorphism ring of 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 (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. It is now not difficult to check that 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.

(Note that the construction does not work for groups that are not abelian: the sum of two homomorphisms need not be a homomorphism in that case.)

We can define the endomorphism ring of any module in exactly the same way; instead of group homomorphisms we then take module homomorphisms of course.

If K is a field and we consider the K-vector space Kⁿ, then its endomorphism ring (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.

In general, endomorphism rings can be defined for the objects of any preadditive category.

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).
• A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotents.