Unique Factorization Monoids and Domains

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring ^[[iW"]] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e., whenever b <a and aMp\bM¿¿0, then aMQbM). By a monoid, we shall always mean a cancellative semigroup with identity element. All rings are assumed to have unities, and integral domains and fields are not assumed to be commutative. If R is an integral domain, then the multiplicative monoid of nonzero elements is denoted by i?x. For any ring or monoid A, U(A) will denote its group of units. A monoid M is called a unique factorization monoid (uf-monoid) iff for every mEM — U(M), any two factorizations of m have a common refinement. That is, if m=aia2 ■ ■ ■ ar = bib2 ■ ■ ■ bs, then the a's and 6's can be factored, say ai = ci • • • c,, a2 = a+i ■ ■ ■ Cj, ■ ■ ■ , 61 = di ■ ■ ■ dk, b2=dk+i ■ ■ • dp, ■ ■ ■ , in such a way that cn=dn for all n. An integral domain R is called a unique factorization domain (uf-domain) iff Rx is a uf-monoid. A monoid M with relation < is said to be ordered by < iff < is a transitive linear ordering such that whenever a<b then ac<bc and ca<cb for all cEM. We call M a positive monoid iff M equals its positive cone M+= {aEM\a^l}. An ordered monoid M is said to be naturally ordered (see [4, p. 154]) iff whenever aM(~\bM¿¿0 and b<a, then aMEbM. It is the purpose of this paper to show ways of constructing ufmonoids and uf-domains. The two principal results are as follows. Theorem 1. The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. Theorem 2. Let M be an ordered monoid and F be a field. The ring F[[M]] of all formal power series with well-ordered support is a ufdomain iff M is naturally ordered. _ Received by the editors April 27, 1970. AMS 1970 subject classifications. Primary 16A02, 20M25; Secondary 06A50.