By Scott T. Chapman
------------------Description-------------------- The research of nonunique factorizations of parts into irreducible components in commutative jewelry and monoids has emerged as an self reliant zone of analysis simply during the last 30 years and has loved a re
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Hence (a) ⊆ (a, b)∗ ∩ (a, c)∗ = ((a, b)(a, c))∗ ⊆ (a); so (a) = ((a, b)(a, c))∗ . So (a, b) and (a, c) are ∗-invertible. Thus (a, b)∗ = (d) and (a, c)∗ = (e) for some d, e ∈ D. So (a) = (d)(e) and hence a = ude for some unit u. But then a ∗-pseudo-irreducible and (d, c)∗ = D gives ud or e is a unit. If ud is a unit, (a) = (e) and c ∈ (a, c)∗ = (e) = (a). Likewise e a unit gives c ∈ (a). So a is ∗-pseudo-prime. (2)=⇒(3) Clear. (3)=⇒(2) Let a be ∗-pseudo-irreducible. Then a has a ∗-comaximal factorization a = a1 · · · an where each ai is ∗-pseudo-prime.
B) P ic(R0 ) = HP ic(R). (c) If R is almost normal, then P ic(R0 ) = P ic(R). Proof. 5]. Our next goal is to determine when Cl(D) = Cl(D[Γ]). 4. 6. Let R be a graded integral domain such that R0 ⊆ R is an inert extension. 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]).
Proof. We have already mentioned part (a). Part (b) is clear by comments in the previous section. 1] and uses facts from . The most popular problem seems to be to determine conditions for A + XB[X] to be an HFD. Partial results have appeared in many of the papers listed above. 3]. Below we give three conditions for A + XB[X] to be an HFD. A surprising consequence of part (b) is that A + XB[X] can be an HFD without B[X] being an HFD. 3. Let A ⊆ B be an extension of integral domains and R = A+XB[X].
Arithmetical Properties of Commutative Rings and Monoids by Scott T. Chapman