##
**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.

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 |

### Keywords:

prime ideals; von Neumann regular rings; generalizations of von Neumann regularity; primitive ideals; right duo rings### Citations:

Zbl 0165.05301; Zbl 0301.16015; Zbl 0355.16017; Zbl 0394.16011; Zbl 0587.16007; Zbl 0836.16005
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\textit{G. F. Birkenmeier} et al., J. Pure Appl. Algebra 115, No. 3, 213--230 (1997; Zbl 0870.16006)

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