If an abelian group has a finite endomorphism ring, then it is necessarily a torsion abelian group . ( Otherwise multiplication by an integer gives an obvious injection from the set of integers into the endomorphism ring . ) In fact, for the same reason, the elements must have bounded order.
32.
If an abelian group has a finite endomorphism ring, then it is necessarily a torsion abelian group . ( Otherwise multiplication by an integer gives an obvious injection from the set of integers into the endomorphism ring . ) In fact, for the same reason, the elements must have bounded order.
33.
Even for group rings, there are examples when the characteristic of the field divides the order of the group : the Jacobson radical of the projective cover of the one-dimensional representation of the alternating group on five points over the field with three elements has the field with three elements as its endomorphism ring.
34.
The Hopf algebroid structure on the endomorphism ring S of the B-bimodule A ( discussed above ) becomes a Hopf algebra in the presence of the hypothesis that the centralizer R = \ { r \ in A : \ forall b \ in B, br = rb \ } is one-dimensional.
35.
The action of the Clifford algebra on ? is defined first by giving the action of an element of " V " on ?, and then showing that this action respects the Clifford relation and so extends to a homomorphism of the full Clifford algebra into the endomorphism ring End ( ? ) by the universal property of Clifford algebras.
36.
Given a right " R "-module U, the set of all " R "-linear maps from " U " to itself forms a ring with addition that is of function and multiplication that is of composition of functions; it is called the endomorphism ring of " U " and is denoted by \ operatorname { End } _ R ( U ).
37.
Conversely, any depth two extension A | B has a Galois theory based on the natural action of \ mbox { End } \, { } _ BA _ B on A : denoting this endomorphism ring by S, one shows S is a left bialgebroid over the centralizer R ( those a in A commuting with all b in B ) with a Galois theory similar to that of Hopf-Galois theory.
38.
There is a right bialgebroid structure on the B-centralized elements T in A \ otimes _ B A dual over R to S; certain endomorphism rings decompose as smash product, such as \ mbox { End } \, A _ B \ cong A \ otimes _ R S, i . e . isomorphic as rings to the smash product of the bialgebroid S ( or its dual ) with the ring A it acts on.
39.
By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable : if and only if the endomorphism ring does not contain an idempotent element different from 0 and 1 . ( If " f " is such an idempotent endomorphism of " M ", then " M " is the direct sum of ker ( " f " ) and im ( " f " ) .)
40.
By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable : if and only if the endomorphism ring does not contain an idempotent element different from 0 and 1 . ( If " f " is such an idempotent endomorphism of " M ", then " M " is the direct sum of ker ( " f " ) and im ( " f " ) .)
