×

Homological characterizations of almost Dedekind domains. (English) Zbl 1440.13057

Let \(R\) be a commutative ring. In this paper, the authors introduce the class of almost projective modules. An R-module \(M\) is said to be almost projective if Ext\(^1_R(M,N) = 0\) for any maximal ideal \(\mathfrak{m}\) and for any \(R_{\mathfrak{m}}\)-module \(N\). It is shown that an \(R\)-module \(M\) satisfying that \(M_{\mathfrak{m}}\) is free over \(R_{\mathfrak{m}}\) for any maximal ideal \(\mathfrak{m}\) of \(R\) is exactly almost projective. Recall that an integral domain \(R\) is almost Dedekind if \(R_{\mathfrak{m}}\) is a discrete valuation domain for each maximal ideal \(\mathfrak{m}\) of \(R\). Some characterizations of almost Dedekind domains \(R\) are given by using the class of almost projective \(R\)-modules, for instance: (1) each ideal of \(R\) is almost projective, (2) each submodule of a projective \(R\)-module is almost projective, … It is also proven that a commutative ring \(R\) is von Neumann regular if and only if each \(R\)-module is almost projective.

MSC:

13C99 Theory of modules and ideals in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, D. D. and Zafrullah, M., Integral domains in which nonzero locally principal ideals are invertible, Commun. Algebra39 (2011) 933-941. · Zbl 1225.13001
[2] Couchot, F., Commutative rings whose cotorsion modules are pure-injective, Palest. J. Math.5 (2016) Spec. Iss., 81-89. · Zbl 1346.13017
[3] Faith, C., Locally perfect commutative rings are those whose modules have maximal submodules, Commun. Algebra23 (1995) 4885-4886. · Zbl 0840.13006
[4] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers, Pure and Applied Mathematics, 90 (Queen’s University, Kingston, 1992). · Zbl 0804.13001
[5] Glaz, S. and Schwarz, R., Prüfer conditions in commutative rings, Arab J. Sci. Eng.36 (2011) 967-983.
[6] Haghany, A., Tolooei, Y. and Vedadi, M. R., Characterizations of commutative max rings and some applications, Indian J. Pure Appl. Math.46 (2015) 371-381. · Zbl 1353.13014
[7] Lam, T. Y., Serre’s Problem on Projective Modules (Springer, Berkeley, 2006). · Zbl 1101.13001
[8] Rotman, J. J., An Introduction to Homological Algebra, 2nd edn. (, Springer, 2009). · Zbl 1157.18001
[9] Uda, H., On a characterization of almost Dedekind domains, Hiroshima Math. J.2 (1972) 339-344. · Zbl 0271.13007
[10] Uda, H., A note on almost Dedekind domains, Mem. Fac. Educ. Miyazaki Univ. (Nat. Sci.)43 (1978) 27-28.
[11] Wang, F. G. and Kim, H., Foundations of Commutative Rings and Their Modules (Singapore, Springer, 2016). · Zbl 1367.13001
[12] Wang, F. G. and Kim, H., Two generalizations of projective modules and their applications, J. Pure Appl. Algebra219 (2015) 2099-2123. · Zbl 1337.13009
[13] Wiegand, R., Descent of projectivity for locally free modules, Proc. Am. Math. Soc.41 (1973) 342-348. · Zbl 0246.13010
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.