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### Zero-Factor Theorem

```
Date: 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.

more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Modern Algebra

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