Thread: composition functions

Hi again :

I have an exercise a little complicated and extensive,




Let be the direct product of copies of 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 be the element of R defined by . Now , if be the element of R defined by . Prove that :



a) (composition functions) is the identity of R ,but is not the identity of R (i.e is a right inverse for , but not a left inverse.
b)Exhibit infinitely many right inverses for
c)Find a nonzero element R such that , but
d)Prove that is no nonzero element p R such that (i,e
is a left zero divisor but not a right zero divisor)



Any help will be well welcome
Thanks

a) you have this written down wrong.

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

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.