# composition functions

• Aug 19th 2012, 07:50 PM
peter1991
composition functions
Hi again :

I have an exercise a little complicated and extensive,

Let $A= \mathbb{Z}\times{}\mathbb{Z}\times{}\mathbb{Z}.... .$ be the direct product of copies of $\mathbb{Z}$ 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 $\emptyset$ be the element of R defined by $\emptyset (a_1,a_2,a_3,......)=(a_2,a_3,........)$. Now , if $\Psi$ be the element of R defined by $\Psi(a_1,a_2,a_3......)=(0,a_1,a_2,a_3........)$ . Prove that :

a) $\emptyset\Psi$(composition functions) is the identity of R ,but $\emptyset\Psi$ is not the identity of R (i.e $\Psi$ is a right inverse for $\emptyset$ , but not a left inverse.
b)Exhibit infinitely many right inverses for $\emptyset$
c)Find a nonzero element $\Pi$ $\in{}$ R such that $\emptyset\Pi=0$ , but $\Pi\emptyset\neq0$
d)Prove that is no nonzero element p $\in{}$ R such that $p\emptyset=0$ (i,e
$\emptyset$ is a left zero divisor but not a right zero divisor)

Any help will be well welcome
Thanks
• Aug 19th 2012, 09:21 PM
Deveno
Re: composition functions
a) you have this written down wrong.

you want to show that θψ = 1R but ψθ ≠ 1R

b) what happens if you replace 0 in the definition of ψ with some other integer k?

c) how about π(a1,a2,a3,....) = (1,0,0,0,......) ?

d) suppose ρθ = 0, that is: ρθ(a1,a2,a3,....) = (0,0,0,....).

this means that ρ(a2,a3,....) = (0,0,0,......), no matter how we choose a2,a3,....

so ρ = 0.