Abstract
We characterize the Puiseux monoids $M$ for which the irreducible and the prime elements in the monoid ring $F[X;M]$, where $F$ is a field, coincide. We present a diagram of implications between some types of Puiseux monoids, with a precise position of the monoids $M$ with this property.
Highlights
The similarities and differences between the ideal theories of monoids and integral domains are studied, for example, in the classical references [3,29,31], as well as in the recent papers [15,16,17].)
If M is a commutative monoid, written additively, and F is a field, the monoid ring F [X; M ] consists of the polynomial expressionsf = a0Xα0 + a1Xα1 + · · · + anXαn, where n ≥ 0, ai ∈ F, and αi ∈ M (i = 0, 1, . . . , n)
If we do not restrict ourselves to submonoids of N0, but consider the submonoids of Q+ = {q ∈ Q : q ≥ 0} instead, for example, the monoid domain F [X; Q+] is an AP domain
Summary
The similarities and differences between the ideal theories of monoids and integral domains are studied, for example, in the classical references [3,29,31], as well as in the recent papers [15,16,17].) We say that a Puiseux monoid M satisfies the gcd/lcm condition if for any t If a Puiseux monoid M satisfies the gcd/lcm condition, for any field F the monoid domain F [X; M ] is AP.
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