Zero-Factor TheoremDate: 02/25/2002 at 09:02:53 From: DeSheng Zhu Subject: Zero-Factor Theorem There is a Zero-Factor Theorem in my textbook, _College Algebra_ (Gustafson/Frisk). If a and b are real numbers, and if ab = 0, then a = 0 or b = 0. Why the restriction 'a and b are real numbers'? Are there any two unreal numbers whose product is zero but where neither itself is zero? Date: 02/25/2002 at 11:55:28 From: Doctor Paul Subject: Re: Zero-Factor Theorem You have to know a bit of modern algebra, but there are algebraic structures in which the product of two nonzero elements is zero. In the ring of integers mod 6, 2*3 = 6 = 0. In the ring of 2x2 matrices, [0 1] [0 1] [0 0] [0 0] = [0 0] [0 0] The idea is that in an Integral Domain the so called zero factor theorem holds. The defining characteristic of an integral domain is that if x and y are both nonzero then xy is nonzero as well. Thus in an integral domain, if xy = 0 then x = 0 or y = 0. Most of the algebraic structures with which you are familiar will be integral domains - the integers and the real numbers are two such examples. But the integers mod 6 and the ring of 2x2 matrices are not integral domains, since I have demonstrated that the product of two nonzero elements can in fact be the zero element. I hope this helps. Please write back if you'd like to talk about this more. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ |
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