When do Commutative Rings with Identity lack the existence of gcd’s?

Question

I’m aware that not all commutative rings guarantee the existence of gcd’s. The only example I have of this currently is the ring of even integers in which 2 has no divisors and therefore no greatest common divisor. I can’t, however, think of a commutative ring with identity in which this is the case. I know that such an example would need to not be a UFD since UFD’s guarantee the existence of gcd’s. Any ideas?