Separators of ideals in multiplicative semigroups of unique factorization domains

摘要

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 ( 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; implies for every . Conversely, if is a congruence on a commutative semigroup C such that the factor semigroup satisfies Condition then there is an ideal I of C such that . Using this result for the multiplicative semigroup of a unique factorization domain D, we show that for every nonzero element , where J(m) denotes the ideal of D generated by m, and is the relation on D defined by if and only if ( is the associate congruence on ). We also show that if a is a nonzero element of a unique factorization domain D then , where d(a) denotes the number of all non associated divisors of a, , and [a] denotes the -class of containing a. As an other application, we show that if d is one of the integers , , , , , , , , then, for every nonzero ideal I of the ring R of all algebraic integers of an imaginary quadratic number field , there is a nonzero element m of R such that P1 = tau(m).