- #1

- 71

- 0

## Homework Statement

True or False?

Let R and S be two isomorphic commutative rings (S=/={0}). Then any ring homomorphism from R to S is an isomorphism.

## Homework Equations

R being a commutative ring means it's an abelian group under addition, and has the following additional properties:

i) a*(b+c)=a*b+a*c

ii) ab=ba

iii) a*(b*c)=(a*b)*c

iv) there exists an element e

_{R}s.t. a*e

_{R}=a for all a in R.

A "ring homomorphism" from R to S is a function f from R to S such that

i) f(a)*f(b)=f(a*b)

ii) f(a+b)=f(a)+f(b)

iii) f(e

_{S})=e

_{R}

## The Attempt at a Solution

BAck of the book says false

I thought to make f(a)=0

_{S}for all a in S which would have worked as a counterexample but but it implies f(e

_{R})=0

_{S}which by property (iii) of ring homomorphisms implies e

_{R}=0

_{S}which means a=a*e

_{S}=a*0

_{S}=0 so a=0 for all a in S but that means S={0} which is a contradiction.

Thanks for reading