Abstract
The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by t-multiplication. This paper investigates integral domains with Boolean t-class semigroup with an emphasis on the GCD and stability conditions. The main results establish t-analogues for well-known results on Prufer domains and Bezout domains of finite character.
Highlights
1 Introduction All rings considered in this paper are integral domains
The class semigroup of a domain R, denoted S(R), is the semigroup of nonzero fractional ideals modulo its subsemigroup of nonzero principal ideals [11,40]
The t-class semigroup of R, denoted St (R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals, that is, the semigroup of the isomorphy classes of the t-ideals of R with the operation induced by ideal t-multiplication
Summary
All rings considered in this paper are integral domains (i.e., commutative with identity and without zerodivisors). A stable domain has Clifford class semigroup [9, Proposition 2.2] and finite character [37, Theorem 3.3]; and an integrally closed stable domain is Prüfer [16, Lemma F]. Of particular relevance to our study is Olberding’s result that an integrally closed domain R is stable if and only if RR is a strongly discrete Prüfer domain of finite character [35, Theorem 4.6].
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