# Thread: RING Ding Dong :))

1. ## RING Ding Dong :))

Given an abelian group $\displaystyle A$. Show that $\displaystyle End(A)$ 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?

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

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