By Scott T. Chapman

------------------Description-------------------- The examine of nonunique factorizations of components into irreducible parts in commutative jewelry and monoids has emerged as an self sustaining quarter of analysis merely during the last 30 years and has loved a re

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Recall that an integral domain D is an independent ring of Krull type if t-Max(D) is independent of ﬁnite character and DP is a valuation domain for each P ∈ t-Max(D). The implication (1)=⇒(3) was ﬁrst given in [31] while the implication (3)=⇒(1) was given in [32]. 8. 8]) For an integral domain D, the following conditions are equivalent. (1 ) D is a semi-rigid GCD domain. (2 ) D is a semi-t-pure GCD domain. (3 ) D is a GCD domain that is an independent ring of Krull type. (4 ) D is an independent ring of Krull type with Clt (D) = 0.

In this section, we investigate to what extent divisibility properties of R are actually determined by the corresponding divisibility properties on the monoid S of nonzero homogeneous elements of R, or by conditions on the homogeneous ideals of R. One can also ask how divisibility properties of R0 relate to those of R. In other words, how do divisibility properties on the submonoids (R0 )∗ and S of R∗ relate to divisibility properties of R (or R∗ )? Let H be a commutative, cancellative (multiplicative) monoid with quotient group H = { a/b | a, b ∈ H }.

Math. Soc. 109 (1990), 907–913. D. Anderson and M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A(7) 8 (1994), 397–402. D. Anderson and M. M. Cohn’s completely primal elements, Zero Dimensional Commutative Rings, 115–123, Lecture Notes in Pure and Appl. , 171, Dekker, New York, 1995. D. Anderson and M. Zafrullah, Independent locally ﬁnite intersections of localizations, Houston J. Math. 25 (1999), 433–452. D. Anderson and M. Zafrullah, Splitting sets in integral domains, Proc. Amer.