By Scott T. Chapman
------------------Description-------------------- The research of nonunique factorizations of parts into irreducible parts in commutative earrings and monoids has emerged as an self sufficient zone of analysis in basic terms over the past 30 years and has loved a re
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Extra info for Arithmetical Properties of Commutative Rings and Monoids
211, Dekker, New York, 1998.  P. Jaﬀard, Les Syst`emes d’Ideaux, Travaux et Recherches Math´ematiques, vol IV, Dunod, Paris, 1960.  P. Malcolmson and F. Okoh, Minimal prime ideals and generalizations of factorial domains, preprint.  E. Matlis, Torsion-free Modules, The University of Chicago Press, ChicagoLondon, 1972.  S. G. Swan, Unique comaximal factorization, J. Algebra (to appear). L. Mott, Convex directed subgroups of the group of divisibility, Canad. J. Math. 26 (1974), 532–542.
Then HCl(R) = Cl(R) if and only if R is almost normal. Proof. 1]. 7. Let R be a Z+ -graded integral domain. Then HCl(R) = Cl(R) if and only if R is almost normal. Proof. 8]. 6 since in this case R0 ⊆ R is an inert extension. 11]. 6. 7]. 2]. In fact, one can deﬁne the divisor class group Cl(Γ) of the Krull monoid Γ, and it turns out that Cl(Γ) ∼ = Cl(K[Γ]) (see  and [54, Section 16]). So it seems natural to ask what happens in the general semigroup ring case. In , it is shown that Cl(D) = Cl(D[G]) when G is a torsionfree abelian group which satisﬁes ACC on cyclic subgroups, and in  it is shown that Cl(D[Γ]) = Cl(D) ⊕ Cl(Γ) when D[Γ] is a PVMD and Γ satisﬁes ACC on cyclic subgroups.
A ﬁrst natural question for polynomial rings is: when does Cl(D) = Cl(D[X]) or P ic(D) = P ic(D[X])? In this case, we have nice answers. 1. Let D be an integral domain. (a) P ic(D) = P ic(D[X]) if and only if D is seminormal. (b) Cl(D) = Cl(D[X]) if and only if D is integrally closed. Proof. 6]. 1 for the Picard group; the class group case will be discussed later. Recall that an integral domain D is called quasinormal if P ic(D) = P ic(D[X, X −1 ]) ( = P ic(D[Z])). An integrally closed domain is always quasinormal, and a quasinormal integral domain is seminormal, but neither converse holds.
Arithmetical Properties of Commutative Rings and Monoids by Scott T. Chapman