In this paper we attempt to generalize the notion of "unique factorization domain" in the spirit of "half-factorial domain". It is shown that this new generalization of UFD implies the now well-known notion of half-factorial domain. As a consequence, we discover that one of the standard axioms for unique factorization domains is slightly redundant.

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Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whether the monoid ring R[x; M ] is atomic provided that both M and R are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for M = N 0 : he constructed an atomic integral domain R such that the polynomial ring R[x] is not atomic. However, the question of whether a monoid algebra F [x; M ] over a field F is atomic provided that M is atomic has been open since then. Here we offer a negative answer to this question. First, we find for any infinite cardinal κ a torsion-free atomic monoid M of rank κ satisfying that the monoid domain R[x; M ] is not atomic for any integral domain R. Then for every n ≥ 2 and for each field F of finite characteristic we exhibit a torsion-free atomic monoid of rank n such that F [x; M ] is not atomic. Finally, we construct a torsion-free atomic monoid M of rank 1 such that Z 2 [x; M ] is not atomic.

In this paper we generalize the standard notion of unique factorization domains to the nonatomic situation. The main result of this paper is that, in contrast to the atomic situation, the assumption that every irreducible is prime (atoms prime) and the notion that every (atomic) nonzero nonunit can be factored uniquely into irreducible elements are distinct notions.