Example: To show that the polynomials of a finite field is itself not a field, I need to show that a multiplicative inverse does not exist.

What are the general techniques that can be used to show that some ring does not have a multiplicative inverse?

I'm thinking the best way to go about it would be to assume that a multiplicative inverse exists by the definition, and then look for a contradiction.

What types of contradictions could help show that a multiplicative inverse doesn't exist?

Or are there any other ideas?