Let be a ring with unity and let be the ring of endomorphisms of . Let , and let be given by .
a) Show that is an endomorphism of
b) Show that is a subring of
c) Prove the analogue of Cayley's theorem for by showing that of (b) is isomorphic to
Let be a ring with unity and let be the ring of endomorphisms of . Let , and let be given by .
a) Show that is an endomorphism of
b) Show that is a subring of
c) Prove the analogue of Cayley's theorem for by showing that of (b) is isomorphic to
(R,+) is an abelian group. so an endomorphism of (R,+) is just a group homomorphism:
so which means is closed under addition.b) Show that is a subring of
so which means is closed under multiplication.
if has an identity element then for all and and thus
for this part you do need to have define the map by then we also havec) Prove the analogue of Cayley's theorem for by showing that of (b) is isomorphic to
so is a ring homomorphism. to show that it's 1-1, let then for all put to get finally, it's trivial that is onto.