On the von Neumann regularity of rings with regular prime factor rings

Fisher, Joe W.; Snider, Robert L.

doi:10.2140/pjm.1974.54.135

Cited by 88 publications

(25 citation statements)

References 24 publications

(16 reference statements)

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“…terminates. It is well known that this definition is left-right symmetric [8] and that strongly ^-regular rings are properly contained in the class of ^-regular rings [10]. THEOREM …”

Section: // R Is Left (Or Right) Quasi-duo and J(r) = 0 Then R Is Rementioning

confidence: 99%

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On quasi-duo rings

1995

Glasgow Math. J.

121

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1. Introduction. Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass' conjecture is true for left duo rings. Call a ring R weakly left duo if for every r e R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rr" {r) is two-sided. Recently, Xue [21] proved that Bass' conjecture is still true for weakly left duo rings.In Section 3 of this paper, we first generalize the above results to left quasi-duo rings. We call a ring R a left (right) quasi-duo ring if every maximal left (right) ideal of R is two-sided. It is shown that the class of all weakly left duo rings is properly contained in the class of all left quasi-duo rings but that several basic properties of weakly left duo rings as proved in [22] are valid for left quasi-duo rings. This is the content of Section 2. Furthermore, we establish in Section 4 the equivalence of the following conditions on a left quasi-duo ring R: (1) R is a left P-exchange ring; (2) Throughout this paper all rings are associative with identity and modules are unitary left modules unless otherwise specified. J(R) always denotes the Jacobson radical of a ring R. Homomorphisms of modules will be written on the side of their arguments opposite to scalars.

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“…terminates. It is well known that this definition is left-right symmetric [8] and that strongly ^-regular rings are properly contained in the class of ^-regular rings [10]. THEOREM …”

Section: // R Is Left (Or Right) Quasi-duo and J(r) = 0 Then R Is Rementioning

confidence: 99%

“…Fisher and Snider [10] proved that a ring R is strongly /r-regular if and only if R/Q is strongly ^-regular for every prime ideal Q of R. (1): Let Q be a prime ideal of R, then R/Q is a division ring by Theorem 3.2 of [11]. So Q is maximal and P(R) = 0 follows from the regularity of R.…”

Section: Let R Be a Right Quasi-duo Ring If Every Prime Ideal Of R Imentioning

confidence: 99%

On quasi-duo rings

1995

Glasgow Math. J.

121

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“…Our proof is of interest because it is quite similar to that of the commutative version given in [16]. R. Snider has given a counterexample to show that the theorem is not true for regular rings (see [10]), and the theorem has been obtained by Fisher and Snider [10] as a corollary of a characterization of regular rings.…”

Section: A Two-sided Quotient Ring S {In Particular the Maximal Two-mentioning

confidence: 66%

Rings of quotients of rings without nilpotent elements

Steinberg¹

1973

Pacific J. Math.

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“…The connection between various generalizations of von Neumann regularity and the condition that every prime ideal is maximal will be investigated. This connection has been investigated by many authors [2,3,5,7,12,14]. The earliest result of this type seems to be by Cohen As a corollary of our main result, we show that if R/P(R) is reduced (i.e., N(R) = P(R) ) then the following are equivalent: (1) R/P(R) is weakly regular; (2) R/¥(R) is right weakly Ti-regular; and (3) every prime ideal of R is maximal.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

A Connection Between Weak Regularity and the Simplicity of Prime Factor Rings

Birkenmeier¹,

Kim²,

Park³

1994

Proceedings of the American Mathematical Society

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Abstract. In this paper, we show that a reduced ring R is weakly regular (i.e., I2 = I for each one-sided ideal / of R ) if and only if every prime ideal is maximal. This result extends several well-known results. Moreover, we provide examples which indicate that further generalization of this result is limited.Throughout this paper R denotes an associative ring with identity. All prime ideals are assumed to be proper. The prime radical of R and the set of nilpotent elements of R are denoted by P(R) and N(R), respectively. The connection between various generalizations of von Neumann regularity and the condition that every prime ideal is maximal will be investigated. This connection has been investigated by many authors [2,3,5,7,12,14]. The earliest result of this type seems to be by Cohen As a corollary of our main result, we show that if R/P(R) is reduced (i.e., N(R) = P(R) ) then the following are equivalent: (1) R/P(R) is weakly regular; (2) R/¥(R) is right weakly Ti-regular; and (3) every prime ideal of R is maximal. This result generalizes Hirano's result for right duo rings. A further consequence of our main result is that if R is reduced then R is weakly regular if and only if every prime factor ring of R is a simple domain. This result can be compared to the well-known fact that when R is reduced, then R is von Neumann regular if and only if every prime factor ring of R is a division ring. We conclude our paper with some examples which illustrate and delimit our results.

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On the von Neumann regularity of rings with regular prime factor rings

Cited by 88 publications

References 24 publications

On quasi-duo rings

On quasi-duo rings

Rings of quotients of rings without nilpotent elements

A Connection Between Weak Regularity and the Simplicity of Prime Factor Rings

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