Separators of Ideals in Multiplicative Semigroups of Unique Factorization Domains

Abstract

In this paper we show that if $I$ is an ideal of a commutative semigroup $C$ such that the separator $SepI$ of $I$ is not empty then the factor semigroup $S=C/P_I$ ($P_I$ is the principal congruence on $C$ defined by $I$) satisfies Condition $(*)$: $S$ is a commutative monoid with a zero; The annihilator $A(s)$ of every non identity element $s$ of $S$ contains a non zero element of $S$; $A(s)=A(t)$ implies $s=t$ for every $s, t\in S$. Conversely, if $\alpha$ is a congruence on a commutative semigroup $C$ such that the factor semigroup $S=C/\alpha$ satisfies Condition $(*)$ then there is an ideal $I$ of $C$ such that $\alpha =P_I$. Using this result for the multiplicative semigroup $D_{mult}$ of a unique factorization domain $D$, we show that $P_{J(m)}=\tau _m$ for every nonzero element $m\in D$, where $J(m)$ denotes the ideal of $D$ generated by $m$, and $\tau _m$ is the relation on $D$ defined by $(a, b)\in \tau _m$ if and only if $gcd(a, m)\sim gcd(b, m)$ ($\sim$ is the associate congruence on $D_{mult}$). We also show that if $a$ is a nonzero element of a unique factorization domain $D$ then $d(a)=|D'/P_{J([a])}|$, where $d(a)$ denotes the number of all non associated divisors of $a$, $D'=D/\sim$, and $[a]$ denotes the $\sim$-class of $D_{mult}$ containing $a$. As an other application, we show that if $d$ is one of the integers $-1$, $-2$, $-3$, $-7$, $-11$, $-19$, $-43$, $-67$, $-163$ then, for every nonzero ideal $I$ of the ring $R$ of all algebraic integers of an imaginary quadratic number field ${\mathbb Q}[\sqrt d]$, there is a nonzero element $m$ of $R$ such that $P_I=\tau _m$.