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Newsgroups: sci.math
From: William Elliot <ma...@rdrop.remove.com>
Date: Sat, 7 Nov 2009 01:20:07 -0800
Local: Sat, Nov 7 2009 8:50 pm
Subject: Re: Newbie Q : Problem from Herstein Regarding Rings
On Fri, 6 Nov 2009, junoexpress wrote: Assume a and b are zero divisors. Thus > "Show that the commutative ring D is an integral domain iff for all > a,b,c in D and a ne 0, ab=ac implies b=c" > I read the logic of this proof as follows: > Let D be a commutative ring and ( D is an integral domain iff for all > I am having trouble with the proof of the back implication of this a,b /= 0; ab = 0; ab = a0; b = 0; 0 /= 0. > PROBLEM: My problem rests with the fact that (in the proof of the back Texts vary as whether a ring has to have unity or not. > implication) we have showed that D is a commutative ring (given by > hypothesis) and that D has no zero divisors, but we have not shown > that D has a unit element, which is the third condition required in > the definition of "integral domain" It seems that the author requires rings to have unity. If he didn't, then he'd have to wright. If R is a communitive ring with unity, then > I went to a few other sources, but they weren't of much help. Before you can jump into a book to pull out theorems, you have > So I don't see how, in problem 10 from Herstein, we can prove that the to know what the author's conventions are: do rings have unity, are maps continuous, are topological spaces Hausdorff, are neighborhoods open? You must Sign in before you can post messages.
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