Ring Isomorphism Help

cs0978

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I seem to be having a difficult time trying to figure out how to do this problem. It's from a non-graded homework assignment. I was able to get every other problem except for this one.

a.) Let R and S be commutative rings and let f: R->S be a ring isomorphism. Prove if x is a zero divisor in R, then f(x) is a zero divisor in S.
b.) Give an example to show that (a) is not necessarily true if R is not isomorphic to S.

Any help would be greatly appreciated! I don't usually do this, but I would really like to understand this topic better.

Thanks!
 

a.) Let R and S be commutative rings and let f: R->S be a ring isomorphism. Prove if x is a zero divisor in R, then f(x) is a zero divisor in S.
b.) Give an example to show that (a) is not necessarily true if R is not isomorphic to S.
Do you know that any isomorphism is a bijection?
Do you know that f(0R)=0S\displaystyle f(0_R)=0_S?

If each of u & v\displaystyle u~\&~v is a zero divisor in R\displaystyle R then 0S=f(0R)=f(uRv)=f(u)Sf(v)\displaystyle 0_S=f(0_R)=f(u\cdot_R v)=f(u)\cdot_S f(v)

What does that tell you?
 
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