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 main congruence of 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 $P_I=\tau _m$.