My Math Forum True/False about Homomorphisms and Zero Divisors

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 May 4th, 2009, 08:04 PM #1 Newbie   Joined: May 2009 Posts: 2 Thanks: 0 True/False about Homomorphisms and Zero Divisors Which are true and false and why: 1) Let f : R --> S be a ring homomorphism, then an element r in R is a zero divisor if and only if f(r) is a zero divisor in S 2) Let f : R --> S be a ring homomorphism that is also surjective. If an element s in S is a zero divisor, then there exists an element r in R such that f(r)=s and r is a zero divisor in R. 3) Let f : R --> S be a ring homomorphism that is also surjective. Then f is a ring isomorphism if and only if the kernel of f contains only the zero element.
 May 5th, 2009, 02:41 AM #2 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: True/False about Homomorphisms and Zero Divisors What have you tried so far? 3 is straightforward... Apply what you know about group homomorphisms. 1 and 2 are really similar.... Think about a really big ring being mapped to a really small ring. Actually... take f : Z --> Z_(n*m) and look at f(n) and f(m). If that isn't enough, show us where you're stuck. Cheers!

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