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### Divisibility of Zero Theory

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Date: 10/06/97 at 14:51:17
From: Anonymous
Subject: Zero .......

Greetings!

I have an adolescent who claims that he has heard of divisibility OF
zero theory at the University. Can you fill me in on this concept and
could you tell me if this is the first time that this idea has been
approached and documented? Also, if you have the time, could you
suggest Web sites that treat the history of this issue?

Thank you,

Ron O'Neal
```

```
Date: 10/06/97 at 16:29:51
From: Doctor Rob
Subject: Re: Zero .......

I think what you are referring to is the idea of zero-divisors. They
are well known in the abstract modern algebra community of
mathematicians, having been around about a century.

There is a mathematical object called a "ring." It has nothing to do
with rings on fingers, bells, or prize fighting! Essentially it is a
set with two operations, addition and multiplication, that behave as
you might expect. If you want lots of details, I can supply them. The
most common example of a ring is the integers (or whole numbers).

A zero-divisor in a ring is an element of the ring which, when
multiplied by some non-zero ring element, gives a product of zero.
The only one in the integers is 0 itself.  In some rings, this is the
only zero-divisor. These rings are very nice, and have many useful and
beautiful properties.

I will give you another example. Consider the set {0,1,2,3}.
To add two of these, add them as integers, and if the result is
bigger than 3, take away 4. Think of a clock with four marks on it,
with 0 up, 1 to the right, 2 down, and 3 to the left, and adding time
on it. This gives the following addition table:

+ | 0 1 2 3
-----------
0 | 0 1 2 3
1 | 1 2 3 0
2 | 2 3 0 1
3 | 3 0 1 2

The element 0 behaves like the zero element, since when you add it to
anything (on either side), you get that thing back. Every element has
a negative:  (-0) = 0, (-1) = 3, (-2) = 2, (-3) = 1.

To multiply two elements of this set, multiply them like integers,
and if the result is bigger than 3, take away 4's until it isn't.
This gives the following multiplication table:

* | 0 1 2 3
-----------
0 | 0 0 0 0
1 | 0 1 2 3
2 | 0 2 0 2
3 | 0 3 2 1

The element 1 behaves like the identity element, since when you
multiply it by anything (on either side), you get that thing back.

This set with these operations is a ring. This ring is called "the
ring of integers modulo four," and is sometimes denoted Z_4, and
sometimes Z/4Z.

Z_4 has nonzero zero-divisors, because 2 is a zero-divisor. In the
above table, you can see that 2*2 = 0, but the second 2 is not 0.
Of course 0 is still a zero-divisor, and you can check that neither
1 nor 3 is.  The set of zero-divisors in Z_4 is {0,2}. This is the
smallest example of a ring which has nonzero zero-divisors.

Many other examples of rings are known, some with nonzero zero-
divisors and some without. Their study is one of the areas of modern
abstract algebra which is covered in upper-level undergraduate and

For history of mathematics, try either or both of these sites:

http://aleph0.clarku.edu/~djoyce/mathhist/mathhist.html

http://www-history.mcs.st-and.ac.uk/~history/

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory
High School Sets

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