Unique factorization monoids and domains

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...

Lectures On Unique Factorization Domains

Unique Factorization in Integral Domains

Throughout R is an integral domain unless otherwise specified. Let A and B be sets. We use the notation A ⊆ B to indicate that A is a subset of B and we use the notation A ⊂ B to mean that A is a proper subset of B. The group of elements in R which have a multiplicative inverse (the group of units of R) is denoted R×. Since R has no zero divisors cancellation holds. If a, b, c ∈ R and a 6= 0 th...

Factorization in Topological Monoids

We sketch a theory of divisibility and factorization in topological monoids, where finite products are replaced by convergent products. The algebraic case can then be viewed as the special case of discretely topologized topological monoids. We define the topological factorization monoid, a generalization of the factorization monoid for algebraic monoids, and show that it is always topologically...

Factorization Properties of Congruence Monoids

Let n ∈ N, Γ ⊆ N and define Γn = {x ∈ Zn | x ∈ Γ} the set of residues of elements of Γ modulo n. If Γn is multiplicatively closed we may define the following submonoid of the naturals: HΓn = {x ∈ N | x = γ, γ ∈ Γn}∪{1} known as a congruence monoid (CM). Unlike the naturals, many CMs enjoy the property of non-unique factorization into irreducibles. This opens the door to the study of arithmetic ...