×

Neat rings. (English) Zbl 1095.13025

A ring is clean if every element is a sum of a unit and an idempotent. A ring is neat if every proper factor of it is clean. The paper has several results on neat rings:
(1) a collection of several equivalent conditions for a commutative ring to be clean.
(2) classification of neatness for some classes of rings:
(a) rings over which every finitely generated module is a direct sum of cyclics,
(b) Bézout domains (in terms of their group of divisibility);
(3) Other examples of neat rings:
(a) every h-local domain;
(b) every domain with Krull dimension.
Besides these, the last two sections of the paper contain generalizations on rigid extension of lattice-ordered abelian groups and relating properties of the additive group of continuous functions of a topological space to its topology. These two sections are independent of the rest of the paper.

MSC:

13F99 Arithmetic rings and other special commutative rings
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, D. D.; Camillo, V. P., Commutative rings whose elements are a sum of a unit and an idempotent, Comm. Algebra, 30, 7, 3327-3336 (2002) · Zbl 1083.13501
[2] Anderson, M.; Feil, T., Lattice Ordered Groups. An Introduction (1988), Reidel: Reidel Dordrecht · Zbl 0636.06008
[3] Banaschewski, B., Gelfand and exchange rings: their spectra in pointfree topology, Arab. J. Sci. Eng., 25, 2C (2000) · Zbl 1271.13052
[4] Brandal, W., Almost maximal integral domains and finitely generated modules, Trans. Amer. Math. Soc., 183, 203-222 (1973) · Zbl 0273.13011
[5] Brandal, W., Commutative Rings Whose Finitely Generated Modules Decompose, Lecture Notes in Mathematics, vol. 723 (1979), Springer: Springer New York · Zbl 0426.13004
[6] Camillo, V. P.; Yu, H., Exchange rings, units and idempotents, Comm. Algebra, 22, 12, 4737-4749 (1994) · Zbl 0811.16002
[7] Conrad, P., Epi-archimedean groups, Czech. Math. J., 24, 192-218 (1974) · Zbl 0319.06009
[8] Conrad, P.; Martinez, J., Complemented lattice-ordered groups, Indag. Math., 1, 3, 281-298 (1990) · Zbl 0735.06006
[9] Contessa, M., On pm-rings, Comm. Algebra, 10, 1, 93-108 (1982) · Zbl 0484.13002
[10] Crawley, P.; Jónnson, B., Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math., 14, 797-855 (1964) · Zbl 0134.25504
[11] Darnel, M., Theory of Lattice-Ordered Groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 187 (1995), Marcel Dekker: Marcel Dekker New York · Zbl 0810.06016
[12] De Marco, G., Projectivity of pure ideals, Rend. Sem. Mat. Univ. Padova, 68, 289-304 (1983) · Zbl 0543.13004
[13] De Marco, G.; Orsatti, A., Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Amer. Math. Soc., 30, 3, 459-466 (1971) · Zbl 0207.05001
[14] Dow, A.; Henriksen, M.; Kopperman, R.; Woods, R. G., Topologies and cotopologies generated by sets of functions, Houston J. Math., 19, 4, 551-586 (1993) · Zbl 0805.54019
[15] Engelking, R., General Topology, Sigma Series in Pure Mathematics, vol. 6 (1989), Heldermann: Heldermann Berlin · Zbl 0684.54001
[16] L. Fuchs, L. Salce, Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001.; L. Fuchs, L. Salce, Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001. · Zbl 0973.13001
[17] L. Gillman, M. Jerison, Rings of Continuous Functions, Graduate Texts in Mathematics, vol. 43, Springer, Berlin, 1960.; L. Gillman, M. Jerison, Rings of Continuous Functions, Graduate Texts in Mathematics, vol. 43, Springer, Berlin, 1960. · Zbl 0093.30001
[18] Gilmer, R., Multiplicative Ideal Theory (1972), Marcel Dekker: Marcel Dekker New York · Zbl 0248.13001
[19] Hager, A. W.; Martinez, J., Fraction-dense algebras and spaces, Canad. J. Math., 45, 5, 977-996 (1993) · Zbl 0795.06017
[20] P.T. Johnstone, Stone Space, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982.; P.T. Johnstone, Stone Space, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. · Zbl 0499.54001
[21] Jøndrup, S., Rings in which pure ideals are generated by idempotents, Math. Scand., 30, 177-185 (1972) · Zbl 0242.13011
[22] Kaplansky, I., Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc., 72, 327-340 (1952) · Zbl 0046.25701
[23] Matlis, E., Torsion-Free Modules, Chicago Lectures in Mathematics (1972), University of Chicago Press: University of Chicago Press Chicago · Zbl 0298.13001
[24] Mott, J. L., Convex directed subgroups of a group of divisibility, Canad. J. Math., 26, 3, 532-542 (1974) · Zbl 0245.06008
[25] Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229, 278-279 (1977) · Zbl 0352.16006
[26] Shutters, W. A., Exchange rings and \(P\)-exchange rings, Notices Amer. Math. Soc., 21 (1974)
[27] A.A. Tuganbaev, Modules with the Exchange Property and Exchange Rings, Handbook of Algebra, vol. 2, North-Holland, Amsterdam, 2000, pp. 439-459.; A.A. Tuganbaev, Modules with the Exchange Property and Exchange Rings, Handbook of Algebra, vol. 2, North-Holland, Amsterdam, 2000, pp. 439-459. · Zbl 1015.16002
[28] Vasconcelos, W. V., Finiteness in projective ideals, J. Algebra, 25, 269-278 (1973) · Zbl 0254.13012
[29] Warfield, R. B., Exchange rings and decompositions of modules, Math. Ann., 199, 31-36 (1972) · Zbl 0228.16012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.