Weakly factorial domains and groups of divisibility

Authors:
D. D. Anderson and Muhammad Zafrullah

Journal:
Proc. Amer. Math. Soc. **109** (1990), 907-913

MSC:
Primary 13F15; Secondary 06F20, 13A05, 13A17, 13G05

DOI:
https://doi.org/10.1090/S0002-9939-1990-1021893-7

MathSciNet review:
1021893

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Abstract: An integral domain is said to be weakly factorial if every nonunit of is a product of primary elements. We give several conditions equivalent to being weakly factorial. For example, we show that the following conditions are equivalent: (1) is weakly factorial; (2) every convex directed subgroup of the group of divisibility of is a cardinal summand; (3) if is a prime ideal of minimal over a proper principal ideal ( ), then has height one and is principal; (4) , where the intersection runs over the height-one primes of , is locally finite, and the -class group of is trivial.

**[1]**D. D. Anderson,*Star-operations induced by overrings*, Comm. Alg.**16**(1988), 2535-2553. MR**955324 (89f:13032)****[2]**D. D. Anderson and L. A. Mahaney,*On primary factorizations*, J. Pure Appl. Alg.**54**(1988), 141-154. MR**963540 (89h:13004)****[3]**A. Bigard, K. Keimal, and S. Wolfenstein,*Groupes et anneaux réticulés*, Lecture Notes in Math., vol. 608, Springer-Verlag, New York, 1977.**[4]**A. Bouvier,*Le groupe des classes d'un anneau integré*, 107 Congrés national des Sociétés savantes, Brest, 1982, sciences, fasc, IV, 85-92.**[5]**A. Bouvier and M. Zafrullah,*On some class groups of an integral domain*, Bull. Soc. Math. Grèce (to appear). MR**1039430 (91g:13015)****[6]**R. Gilmer,*Multiplicative ideal theory*, Marcel Dekker, New York, 1972. MR**0427289 (55:323)****[7]**M. Griffin,*Some results on**-multiplication rings*, Canad. J. Math.**19**(1967), 710-722. MR**0215830 (35:6665)****[8]**I. Kaplansky,*Commutative rings*, revised ed., The University of Chicago Press, Chicago and London, 1974. MR**0345945 (49:10674)****[9]**J. L. Mott,*The group of divisibility and its application*, Lecture Notes in Math., vol. 311, Springer-Verlag, New York, 1972, 194-208. MR**0337943 (49:2712)****[10]**-,*Convex directed subgroups of a group of divisibility*, Canad. J. Math.**26**(1974), 532-542. MR**0364213 (51:468)****[11]**J. L. Mott and M. Schexnayder,*Exact sequences of semi-value groups*, J. Reine Angew. Math.**283/284**(1976), 388-401. MR**0404247 (53:8050)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1021893-7

Article copyright:
© Copyright 1990
American Mathematical Society