Irreducible elements in commutative rings with zero-divisors

Let R be a commutative ring with zero-divisors. A nonunit element a in R is irreducible if a=bc implies (a)=(b) or (a) =(c). We show that if a,b in R with a irreducible and (a)? (b) ?R then a is a zero-divisor and b is a non zero-divisor. It follows that a in R is irreducible if and only if (1) (a) is maximal in the set of proper principal ideals of R or (2) (a) is maximal in the set of principal ideals generated by zero-divisors. Thus a chain (a1) ?...(an) of principal ideals generated by irreducible elements must have n ? 2.