# Ring, Integral Domain

• May 15th 2011, 02:06 PM
Ring, Integral Domain
If R is a ring and a\neq 0\in R and ab=ca, then b=c. How can I show that R is an integral domain?
• May 15th 2011, 02:11 PM
tonio
Quote:

Originally Posted by road
If R is a ring and a\neq 0\in R and ab=ca, then b=c. How can I show that R is an integral domain?

Suppose $a\neq 0\,\,and\,\,a\cdot b=0=a\cdot 0$ . and now use the given data.

Tonio
• May 15th 2011, 02:12 PM
TheEmptySet
Quote:

Originally Posted by road
If R is a ring and a\neq 0\in R and ab=ca, then b=c. How can I show that R is an integral domain?

As written the above statement is not true unless R is a commutative ring.