# Integral domain

1. Dec 3, 2008

### fk378

1. The problem statement, all variables and given/known data
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.

3. The attempt at a solution
I don't know where to begin with this.

2. Dec 3, 2008

### mutton

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

3. Dec 3, 2008

### 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?

4. Dec 3, 2008

### 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.

5. Dec 3, 2008

### pjm372

I think Michael would be ashamed