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Math Help - RING Ding Dong :))

  1. #1
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    Red face RING Ding Dong :))

    Given an abelian group A. Show that End(A) 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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  2. #2
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    Quote Originally Posted by GTK X Hunter View Post
    Given an abelian group A. Show that End(A) 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?
    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 \text{End}(\mathbb{Z}) is commutative because it's isomorphic to \mathbb{Z}. but R=\text{End}(\mathbb{Z} \oplus \mathbb{Z}) is not commutative. here is why: define f,g \in R by f(m,n)=(n,m) and

    g(m,n)=(m+n,n), for all m,n \in \mathbb{Z}. then fg(m,n)=f(m+n,n)=(n,m+n) but gf(m,n)=g(n,m)=(m+n,m). so fg \neq gf and hence R is not commutative.
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