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

By Scott T. Chapman

------------------Description-------------------- The learn of nonunique factorizations of components into irreducible components in commutative jewelry and monoids has emerged as an self reliant zone of analysis basically during the last 30 years and has loved a re

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Dumitrescu, Condensed domains, Canad. Math. Bull. 46 (2003), 3–13. [9] D. D. A. Mahaney, On primary factorizations, J. Pure Appl. Algebra 54 (1988), 141–154. D. Anderson, J. L. Mott, and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Mat. Ital. B(7) 6 (1992), 613–630. D. L. Mott, and M. Zafrullah, Unique factorization in nonatomic integral domains, Boll. Unione Mat. Ital. Sez B Artic Ric. Mat. (8) 2 (1999), 341–352. D. O. Quinterro, Some generalizations of GCD-domains, Factorization in Integral Domains, 189–195, Lecture Notes in Pure and Appl.

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. Then ρ(D[Γ]) = D(Cl(K[Γ]))/2. Proof. 3]. 5 The Picard Group and Class Group of a Graded Domain In this section, we investigate the Picard group and (t-)class group of a graded integral domain.

Now up to associates the nonunits (atoms) of D have the form uX n where u ∈ L∗ and n ≥ 1 (n = 1). Let u, v ∈ L∗ . Then uX and vX are associates ⇐⇒ uv −1 ∈ K and for n > 1, uX n = (vX)(uv −1 X n−1 ). Hence [uX n1 , vX n2 ] = 1 ⇐⇒ n1 = n2 = 1 and uv −1 ∈ / K. Let a be a nonzero nonunit of D. Then either a is [ ]-pseudo-irreducible or a = a1 · · · an where each ai is a nonunit and [ai , aj ] = 1 for i = j. In the latter case [ai , aj ] = 1 gives that each ai has order one and hence is irreducible and thus [ ]-pseudo-irreducible.

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