It Cuts Both Ways

I've been having this nagging doubt that I've forgotten more math than I know and all the rest of the mathematicians would laugh at me if they knew of my gaping ignorance. Last night I was talking to a topologist (a very successful topologist, I might add) that I know from grad school. (For those who are playing along at home, yes, I am speaking of the person referenced herein as Timothy, with all that entails.) Topologist is looking at polynomials in Z_4[x] and was asking me questions about its ring structure.

He said he noticed that the ring is not a UFD. He asked me if it's a PID. (This I knew.)

He asked something about the irreducible polynomials. I told him that if you modded out by the ideal generated by an irreducible, then the quotient is an integral domain. He said, "Really?" (OK, at this point I wasn't 100% sure since prime matrix rings usually aren't domains, but since Z_4[x] is commutative...)

I mentioned something about the zero divisors; he said, "It has zero divisors?"

So I was pleased to learn that I am not the only person who has forgotten stuff. But then he asked a difficult question:

Him: Remember the course on group cohomology that we took from Professor Walker?

Me: We didn't take a course on group cohomology.

Him: We did. Chip and Owen were in the class.

Me: We never took a class together with Chip in it. Chip was in Walker's K-Theory class that I took.

Him: We used Rotman's book.

Me: We used Rotman in K-Theory. With the vector bundles?

Him: Vector bundles are K-Theory. I'm talking about group cohomology.

Me: We took a course on group cohomology? Are you sure?