Proving Integral Domain of D Using Commutative Ring

[fk378]

Homework Statement

Given a,b,c in D, with a not 0, we have ab=ac implies b=c. Show that the commutative ring D is an integral domain.

The Attempt at a Solution

I don't know where to begin with this.

 

[mutton]

You want to show that there are no 0 divisors. Look at what's given. Look at its contrapositive.

 

[fk378]

So we want to assume it is not an integral domain, then show that b does not equal c?

Well, I know it is possible for b to not equal c because if we are in say, Z mod 6, then [0]=[3]=[6]. But how do I generalize this? Is this the right method to go about it?

 

[mutton]

You don't have to prove by contradiction. What I meant was that since we have

a \ne 0, ab = ac implies b = c,

we also know

a \ne 0, b \ne c implies ab \ne ac. Letting b = 0 or c = 0 should get you what you want.

 

[pjm372]

I think Michael would be ashamed