Divisibility of Zero Theory

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 
lower-level graduate courses.

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/   

I hope that this answers your question. If not, write again.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/