Given an abelian group . Show that under homomorphism addition and homomorphism multiplication operation which is defined as composition of function, is a ring. Does this ring have unity element? Is this ring commutative?

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- December 23rd 2009, 10:34 AMGTK X HunterRING Ding Dong :))
Given an abelian group . Show that under homomorphism addition and homomorphism multiplication operation which is defined as composition of function, is a ring. Does this ring have unity element? Is this ring commutative?

- December 23rd 2009, 07:29 PMNonCommAlg
the first part of your question is straightforward and so is left for you. the answer to the second part is yes. the identity map is the unity element. the third part is interesting! the answer

is "not necessarily". for example is commutative because it's isomorphic to but is not commutative. here is why: define by and

for all then but so and hence is not commutative.