×

Regularity conditions and the simplicity of prime factor rings. (English) Zbl 0870.16006

J. Pure Appl. Algebra 115, No. 3, 213-230 (1997); corrigendum ibid. 149, No. 1, 105 (2000).
A ring \(R\) is said to satisfy the pm-condition if every prime ideal of \(R\) is maximal, or equivalently, all prime factor rings of \(R\) are simple. A ring \(R\) is said to be von Neumann regular if for every \(x\in R\), there exists \(y\in R\) such that \(xyx=x\). The class of von Neumann regular rings is a very important and interesting class of rings in ring theory, and this notion of regularity has been generalized to many other weaker but related regularities (e.g. \(\pi\)-regular, strongly \(\pi\)-regular, weakly \(\pi\)-regular, weakly regular, bi-regular, etc.).
The connections between the pm-condition and various generalizations of von Neumann regularity have been interesting to many researchers over the past three decades. The earliest result of this type was proved by H. H. Storrer [Comment. Math. Helv. 43, 378-401 (1968; Zbl 0165.05301)]: For a commutative ring \(R\), the following conditions are equivalent: (1) \(R\) is \(\pi\)-regular; (2) \(R/P(R)\) is von Neumann regular; (3) Every prime ideal of \(R\) is maximal. J. W. Fisher and R. L. Snider extended this result to P.I. rings [Pac. J. Math. 54, 135-144 (1974; Zbl 0301.16015), Theorem 2.3]. In another direction, V. R. Chandran [Indian J. Pure Appl. Math. 8, No. 1, 54-59 (1977; Zbl 0355.16017), Theorem 3] generalized Storrer’s result to duo rings (i.e. every one-sided ideal is two-sided). Later, Chandran’s result was further generalized to right duo rings (every right ideal is two-sided) by Y. Hirano [Math. J. Okayama Univ. 20, 141-149 (1978; Zbl 0394.16011), Corollary 1], and to bounded weakly right duo rings (see below for definition) by X. Yao [Pure Appl. Math. Sci. 21, 19-24 (1985; Zbl 0587.16007), Theorem 3]. Recently, the reviewer proved that for a right quasi-duo ring \(R\), if every prime ideal is right primitive, then \(R\) is strongly \(\pi\)-regular and \(R/J(R)\) is strongly regular [H.-P. Yu, Bull. Aust. Math. Soc. 51, No. 3, 433-437 (1995; Zbl 0836.16005), Theorem 2.5].
The paper under review is trying to unify and extend many of the results in the study of this connection, by investigating the two general questions that seem to underline the papers mentioned above: 1. What is the connection between the various generalizations of regularity and the conditions that all or some subclasses of prime ideals are primitive or maximal? 2. For a ring \(R\), when can some type of regularity condition on \(R/I\) be lifted to \(R\), where \(I\) is a certain type of ideal of \(R\).
The main results are Theorems 2.6 and 2.8 in the paper (which are too technical to be stated here in a short review). Also included are some nice constructions and examples that show the proper extension of the main results, and delimit future possible generalizations.
Reviewer: H.-P.Yu (Emory)

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D25 Ideals in associative algebras
16N60 Prime and semiprime associative rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baer, R., Radical ideals, Amer. J. Math., 65, 537-568 (1943) · Zbl 0060.07104
[2] Beidar, K.; Wisbauer, R., Properly semiprime self-pp-modules, Comm. Algebra, 23, 841-861 (1995) · Zbl 0822.16017
[3] Belluce, L. P., Spectral spaces and non-commutative rings, Comm. Algebra, 19, 1855-1865 (1991) · Zbl 0728.16002
[4] Birkenmeier, G. F., A survey of regularity conditions and the simplicity of prime factor rings, Vietnam J. Math., 23, 1, 29-38 (1995) · Zbl 1038.16009
[5] Birkenmeier, G. F.; Heatherly, H. E.; Lee, Enoch K., Completely prime ideals and associated radicals, (Jain, S. K.; Rizvi, S. T., Proc. Biennial Ohio State-Denison Conf.. Proc. Biennial Ohio State-Denison Conf., 1992 (1993), World Scientific: World Scientific New Jersey), 102-129 · Zbl 0853.16022
[6] Birkenmeier, G. F.; Kim, J. Y.; Park, J. K., A connection between weak regularity and the simplicity of prime factor rings, (Proc. Amer. Math. Soc., 122 (1994)), 53-58 · Zbl 0814.16001
[7] Camillo, V.; Xiao, Y., Weakly regular rings, Comm. Algebra, 22, 4095-4112 (1994) · Zbl 0810.16011
[8] Chandran, V. R., On two analogues of Cohen’s theorem, Indian J. Pure Appl. Math., 8, 54-59 (1977) · Zbl 0355.16017
[9] Cohen, I. S., Commutative rings with restricted minimum conditions, Duke Math. J., 17, 27-42 (1950) · Zbl 0041.36408
[10] Cozzens, J.; Faith, C., Simple Noetherian Rings (1975), Cambridge Univ. Press: Cambridge Univ. Press London · Zbl 0314.16001
[11] Dischinger, F., Sur les anneaux fortment π-reguliers, C.R. Acad. Sci. Paris Ser. A-B, 283, 571-573 (1976) · Zbl 0338.16001
[12] Fisher, J. W.; Snider, R. L., On the von Neumann regularity of rings with regular prime factor rings, Pacific J. Math., 54, 135-144 (1974) · Zbl 0301.16015
[13] Gupta, V., Weakly π-regular rings and group rings, Math. J. Okayama Univ., 19, 123-127 (1977) · Zbl 0371.16009
[14] Gupta, V., A generalization of strongly regular rings, Acta Math. Hung., 43, 57-61 (1984) · Zbl 0535.16015
[15] Hirano, Y., Some studies on strongly π-regular rings, Math. J. Okayama Univ., 20, 141-149 (1978) · Zbl 0394.16011
[16] Jacobson, N., Structure of Rings, (Amer. Math. Soc. Colloq. Publ., Vol. 37 (1968)), Providence
[17] Ramamurthi, V. S., Weakly regular rings, Canad. Math. Bull., 16, 317-321 (1973) · Zbl 0241.16007
[18] Rowen, L. H., Ring Theory I (1988), Academic Press: Academic Press San Diego · Zbl 0651.16001
[19] Shin, G., Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc., 184, 43-60 (1973) · Zbl 0283.16021
[20] Stock, J., On rings whose projective modules have the exchange property, J. Algebra, 103, 437-453 (1986) · Zbl 0603.16016
[21] Storrer, H. H., Epimorphismen von kommutativen Ringen, Comment Math. Helv., 43, 378-401 (1968) · Zbl 0165.05301
[22] Sun, S.-H., Noncommutative rings in which every prime ideal is contained in a unique maximal ideal, J. Pure Appl. Algebra, 76, 179-192 (1991) · Zbl 0747.16001
[23] Sun, S.-H., On biregular rings and their duality, J. Pure Appl. Algebra, 89, 329-337 (1993) · Zbl 0801.16032
[24] Xue, Yao, Weakly right duo rings, Pure Appl. Math. Sci., 21, 19-24 (1985) · Zbl 0587.16007
[25] Hua-Ping, Yu, On quasi-duo rings, Glasgow Math. J., 37, 21-31 (1995) · Zbl 0819.16001
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.