A Dedekind ring is a commutative ring in which the following hold.
1. It is a noetherian ring and an integral domain.
2. It is integrally closed in its field of fractions.
3. Every nonzero prime ideal is also a maximal ideal. In any commutative ring with multiplicative identity, maximal ideals are always prime ideals.
The main example of a Dedekind domain is the ring of algebraic integers in a number field, an extension field of the rational numbers. An important consequence of the above axioms is that every nonzero ideal can be written uniquely, up to ordering, as a product of nonzero prime ideals, with the unit ideal represented by the empty product. This compensates for the possible failure of unique factorization of elements into irreducibles.
Grams (1974, pp. 437-439) constructed a Dedekind domain 
such that, for every integer 
, there is a nonzero element 
whose principal ideal 
has exactly two distinct prime
ideal divisors, although 
has at least 
irreducible divisors
counted up to multiplication by a ring
unit. Equivalently, 
is not a ring unit in exactly two of the valuation
rings obtained by localizing 
at its nonzero prime
ideals.
Grams (1974, p. 437) used this construction to refute the bound suggested by the hint to Exercise 25(a) of Bourbaki's Algèbre commutative.
The hint would bound the number of such irreducible
divisors by the number of valuation rings in which
is not a ring
unit, which here is two. The exercise itself asserts only finiteness of the number
of such divisors for each nonzero element
of an intersection of valuation
rings, assuming that every nonzero element
of their common field of fractions is a ring
unit in all but finitely many of those rings. She explicitly
stated that she knew no counterexample to the exercise itself.
