Hi again :

I have an exercise a little complicated and extensive,

Let $\displaystyle A= \mathbb{Z}\times{}\mathbb{Z}\times{}\mathbb{Z}.... . $ be the direct product of copies of $\displaystyle \mathbb{Z}$ indexed with the positive integers (so A is a ring under componentwise addition and multiplication) and let R be the ring of all group homomorphism from A to itself with addition defined as pointwise addition of functions . If $\displaystyle \emptyset$ be the element of R defined by $\displaystyle \emptyset (a_1,a_2,a_3,......)=(a_2,a_3,........)$. Now , if $\displaystyle \Psi$ be the element of R defined by $\displaystyle \Psi(a_1,a_2,a_3......)=(0,a_1,a_2,a_3........)$ . Prove that :

a)$\displaystyle \emptyset\Psi$(composition functions) is the identity of R ,but $\displaystyle \emptyset\Psi$ is not the identity of R (i.e $\displaystyle \Psi$ is a right inverse for $\displaystyle \emptyset$ , but not a left inverse.

b)Exhibit infinitely many right inverses for $\displaystyle \emptyset$

c)Find a nonzero element $\displaystyle \Pi$ $\displaystyle \in{}$ R such that $\displaystyle \emptyset\Pi=0$ , but $\displaystyle \Pi\emptyset\neq0$

d)Prove that is no nonzero element p $\displaystyle \in{}$ R such that $\displaystyle p\emptyset=0$ (i,e

$\displaystyle \emptyset$ is a left zero divisor but not a right zero divisor)

Any help will be well welcome

Thanks