Abstract algebra: Rings and Ideals

[lmedin02]

Homework Statement

The problem is to show that a subset A of a ring S is an ideal where A has certain properties. S is a ring described as a cartisian product of two other rings (i.e., S=(RxZ,+,*)). I have already proved that A is a subring of S and proved one direction of the definition of an ideal. But, the other direction has brought me to having to show that R is commutative. It is given that R is a ring without zero divisors and without identity.

Homework Equations

The Attempt at a Solution

I know that a ring R is commutative if it has the property that ab=ca implies b=c when a is not zero. I have attempted various simple manipulations of this statement by using the fact that R is a ring without zero divisors and without an identity.

 

[lmedin02]

In other words, is a ring R without zero divisors and without an identity commutative.

 

[VeeEight]

In other words, is a ring R without zero divisors and without an identity commutative.

No, see domain: http://en.wikipedia.org/wiki/Domain_(ring_theory)

 

[lmedin02]

An integral domain has a unity (i.e., identity). In my case, R has no unity so it is not an integral domain.