Let k be a field and X an indeterminate over k . In this note we prove that the domain kwhere p, q, r are positive integers is not. We also prove that kif and only if q is even. These are very special cases of well-known results on semigroup rings, but our proofs are mainly concerned with the computation of the dual (equivalently the inverse) of the maximal ideal of the ring.

The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal tmultiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prüfer v-multiplication domain (PVMD) is of Krull type (in the sense of Griffin [24]) if and only if its t-class semigroup is Clifford. This extends Bazzoni and Salce's results on valuation domains [11] and Prüfer domains [7,8,9,10].S. Kabbaj and A. Mimouni, t-Class semigroups the problem for the class of integrally closed domains by proving that "R is an integrally closed domain with Clifford class semigroup if and only if R is a Prüfer domain of finite character" [10], Theorem 4.5. It is worth recalling that, in the series of papers [39,40,41], Olberding undertook an extensive study of (Lipman and Sally-Vasconcelos) stability conditions which prepared the ground to address the correlation between stability and the theory of class semigroups.AIdeal t-multiplication converts ring notions such as PID, Dedekind, Bézout (of finite character), Prüfer (of finite character), and integrality to UFD, Krull, GCD (of finite t-character), PVMD (of finite t-character), and pseudo-integrality, respectively. Recall at this point that the PVMDs of finite t-character (i.e., each proper t-ideal is contained in only finitely many t-maximal ideals) are exactly the Krull-type rings introduced and studied by Griffin in 1967-68 [23, 24]. Also pseudo-integrality (which should be termed t-integrality) was introduced and studied in 1991 by D. F. Anderson, Houston and Zafrullah [4]. We'll provide more details about this property which turned to be crucial for our study.This paper examines ring-theoretic properties of an integral domain which reciprocally reflect in semigroup-theoretic properties of its t-class semigroup. Notions and contexts that suit best t-multiplication are studied in an attempt to parallel analogous developments and generalize well-known results on class semigroups. Recall from [10,32] that an integral domain R is Clifford regular (resp., Boole regular) if S(R) is a Clifford (resp., Boolean) semigroup. A first correlation between regularity and stability conditions can be sought through Lipman stability. Indeed, R is called an L-stable domain if n≥1 (I n : I n ) = (I : I) for every nonzero ideal I of R [1]. Lipman introduced the notion of stability in the specific setting of one-dimensional commutative semi-local Noetherian rings in order to give a characterization of Arf rings; in this context, L-stability coincides with Boole regularity [37]. By analogy, we call an integral domain R Clifford (resp., Boole) t-regular if S t (R) is a Clifford (resp., Boolean) semigroup. Clearly, a Boole t-regular domain is Clifford t-re...

This paper seeks ring-theoretic conditions of an integral domain R that reflect in the Clifford property or Boolean property of its class semigroup S(R), that is, the semigroup of the isomorphy classes of the nonzero (integral) ideals of R with the operation induced by multiplication. Precisely, in Section 3, we characterize integrally closed domains with Boolean class semigoup; in this case, S(R) identifies with the Boolean semigroup formed of all fractional overrings of R. In Section 4, we investigate Noetherian-like settings where the Clifford and Boolean properties of S(R) coincide with (Lipman and Sally-Vasconcelos) stability conditions; a main feature is that the Clifford property forces t−locally Noetherian domains to be one-dimensional Noetherian domains. Section 5 studies the transfer of the Clifford and Boolean properties to various pullback constructions. Our results lead to new families of integral domains with Clifford or Boolean class semigroup, moving therefore beyond the contexts of integrally closed domains or Noetherian domains.

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