1. The problem statement, all variables and given/known data Let a, b be members of a commutative ring with identity R. If a is a a prime and a, b are associates then b is also prime. True/False 2. Relevant equations Definitions: a is prime if a|xy implies a|x or a|y a and b are associates if there exists a unit u s.t a=bu 3. The attempt at a solution First off I'm not sure whether its even true or not; so I've tried to both prove it and find a counter example: Attempt at a proof: let b|xy. Then if b = au for some unit u, au|xy, that is for some k in R auk = xy. so ak divides xy(u^(-1)) but k does not necessarily have an inverse so this doesn't really get me anywhere. I've shown in an earlier question that if a is irreducible and a, b are associates then b is irreducible. Prime implies irreducible but not vice versa so if there is a counter example it will depend on an element of R that is irreducible but not prime. I know examples of these can be found in rings such as Z(5i) (the set of numbers a+5bi for a, b in Z). But I don't know how to find units of this ring. Consider (a+5bi)(c+5di) = 1. This gives the simultaneous equations 1+25bd-ac = 0 = ad+bc But so far I have not managed to find any integer solutions (and to be honest I don't know any way of doing this other than trial and error). Thanks for any help!