By Scott T. Chapman

ISBN-10: 0824723279

ISBN-13: 9780824723279

ISBN-10: 1420028243

ISBN-13: 9781420028249

------------------Description-------------------- The examine of nonunique factorizations of parts into irreducible components in commutative earrings and monoids has emerged as an self sufficient zone of analysis in basic terms during the last 30 years and has loved a re

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4]. 4]. 2]. For more on HFDs, see the survey article [39]. Several divisibility properties are studied for group rings and semigroup rings in [54] and [70]. We mention three for group rings. 1. Let D be an integral domain and G a torsionfree abelian group. , is a BFD, FFD) and G satisﬁes ACC on cyclic subgroups. Proof. 1], respectively. We next consider Z+ -graded integral domains of the form R = ⊕n∈Z+ Dn X n , where {Dn }n∈Z+ is a family of subrings of an integral domain D with Dn ⊆ Dn+1 for all n ∈ Z+ .

For other related results, see [28], [31], [36], [63], and [72] for semigroup rings, and [35], [58], [59], [60], [61], and [72] for the A + XB[X] and related constructions. 5. (a) Let K be a ﬁeld and R = K[X n , X n+1 , . . , X 2n−1 ]. Then ρ(R) = (nD(P ic(R)) + 3n − 2)/2n. (b) For n ≥ 2, let K0 ⊆ K1 ⊆ . . ⊆ Kn−1 be an ascending sequence of subﬁelds of a ﬁeld K with Kn−1 K, and let R = K0 + K1 X + · · · + Kn−1 X n−1 + X n K[X]. Then ρ(R) = (2n + D(P ic(R)) − 1)/2. (c) Let D be a UFD with proper quotient ﬁeld K and D[Γ] a Krull domain which is not a UFD.

We will see that things usually behave better when R0 ⊆ R is an inert extension. An integral ideal I of R = ⊕α∈Γ Rα is homogeneous if I = ⊕α∈Γ (I ∩ Rα ); equivalently, if I is generated by homogeneous elements. A fractional ideal I of R is homogeneous if sI is an integral homogeneous ideal of R for some s ∈ S (thus I ⊆ RS ). For x = x1 + · · · + xn ∈ RS with each xi ∈ (RS )αi and α1 < . . < αn , we deﬁne the content of x to be the homogeneous ideal C(x) = (x1 , . . , xn ). Thus a fractional ideal I ⊆ RS is homogeneous if and only if C(x) ⊆ I for each x ∈ I.

### Arithmetical Properties of Commutative Rings and Monoids by Scott T. Chapman

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