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**1. Homework Statement**

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. Homework 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!