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This paper studies the class group of a graded integral domain R = ⊕ α∈Γ R α . We prove that if the extension R 0 ⊂ R is inert, then Cl(R) = HCl(R) if and only if R is almost normal. As an application, we state a decomposition theorem for class groups of semigroup rings, namely, Cl(A[Γ]) ∼ = Cl(A) ⊕ HCl(K[Γ]) if and only if A[Γ] is integrally closed. This recovers the well-known results developed for the classic contexts of polynomial rings and Krull semigroup rings. Further, we obtain an interesting result on the natural homomorphism φ : Cl(A) → Cl(A[Γ]), that is, Cl(A[Γ]) = Cl(A) if and only if A and Γ are integrally closed and Cl(Γ) = 0. Our results are backed by original examples. MSC: 13C20; 13A02; 13F05; 13B22 Let R be an integral domain. Following [6], we define the class group of R, denoted Cl(R), to be the group of t-invertible fractional t-ideals of R under t-multiplication modulo its subgroup of principal fractional ideals. Divisibility properties of a domain R are often reflected in group-theoretic properties of Cl(R). For R a Krull domain, Cl(R) is the usual divisor class group of R. In this case, Cl(R) = 0 if and only if R is factorial. If R is a Prüfer domain, then Cl(R) = P ic(R) is the ideal class group of R. In this case, Cl(R) = 0 if and only if R is a Bezout domain. We assume familiarity with class groups and related concepts, as in [6], [10], and [12]. On one hand, a well-known result is that if R is a Z + -graded Krull domain, then Cl(R) is generated by the classes of homogeneous height-one prime ideals of R [10, Proposition 10.2], i.e., Cl(R) = HCl(R), where HCl(R) is the homogeneous class group of R. In [3, Theorem 4.2], D.F. Anderson showed that the same holds for any Γ-graded Krull domain, where Γ is an arbitrary torsionless grading monoid. So one may remove the "Krull assumption" and legitimately ask the following: For an arbitrary graded domain R, how does the equality "Cl(R) = HCl(R)" reflect in ring-theoretic properties of R? On the other hand, many authors investigated the problem of characterizing ring-theoretic properties in terms of Picard groups. In [7], Bass and Murthy proved that for an integral domain A, if P ic(A[X, X −1 ]) = P ic(A) then A is seminormal. However, Pedrini showed that the converse fails to be true in general [22, p. 96]. Many years later, Gilmer and Heitmann [14] solved completely the problem of characterizing "seminormality" in terms of Picard groups. They stated that P ic(A[X]) = P ic(A) if and only if A is seminormal. In the same line, in 1982, the Andersons [1] examined the property of almost normality for graded domains. They established that if R is an almost normal graded domain with R 0 ⊂ R inert, then P ic(R) = HP ic(R). So the problem remained somehow open. However, in 1987, Gabelli proved that for an integral domain A, Cl(A[X]) = Cl(A) if and only if A is integrally closed [11, Theorem 3.6]. Recall for convenience that A[X], graded in the natural way, is almost normal if and only if A[X] (and hence A) is integrally closed. This motivates our second...

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Let D be an integral domain with quotient field K. A Bhargava ring over D is defined to be x D = f ∈ K X ∀a ∈ D f xX + a ∈ D X , where x ∈ D. A Bhargava ring over D is a subring of the ring of integer-valued polynomials over D. In this article, we study the prime ideal structure and calculate the Krull and valuative dimension of Bhargava rings over a general domain D.