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1.
In this paper, we study rings with the annihilator condition (a.c.) and rings whose space of minimal prime ideals, , is compact. We begin by extending the definition of (a.c.) to noncommutative rings. We then show that several extensions over semiprime rings have (a.c.). Moreover, we investigate the annihilator condition under the formation of matrix rings and classical quotient rings. Finally, we prove that if R is a reduced ring then: the classical right quotient ring Q(R) is strongly regular if and only if R has a Property (A) and is compact, if and only if R has (a.c.) and is compact. This extends several results about commutative rings with (a.c.) to the noncommutative setting. 相似文献
2.
L.J. Ratliff 《Journal of Pure and Applied Algebra》1978,12(2):159-175
A chain condition intermediate to the catenary property and the chain condition for prime ideals (c.c.) is studied. Like the c.c., the condition is inherited from a semi-local domain R by integral extension domains, by local quotient domains, and by factor domains, and a semi-local ring that satisfies the condition is catenary. (Unlike the c.c., none of these statements is true when R is not semi-local.) A number of characterizations of a semi-local domain that satisfies the condition are given in terms of: integral (respectively, algebraic, transcendental) extension domains, Henselizations, completions, Rees rings, associated graded rings and certain discrete valuation over-rings. Then four of the catenary chain conjectures are characterized in terms of this condition. 相似文献
3.
We study the spectrum of minimal prime ideals of commutative Bezout rings. We apply the results obtained to the problem of
diagonal reduction of matrices over rings of this sort.
Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 1001–1005, July, 1999. 相似文献
4.
Louis Rowen 《Israel Journal of Mathematics》1977,27(2):131-149
The study of pivotal monomials (and related conditions) is continued and extended, with the aim of studying carefully a situation
generalizing Martindale's theory of prime rings with generalized polynomial identity. This is used to describe various classes
of rings in terms of simple elementary sentences. The focus is on prime “Johnson” rings, which play a crucial role in our
characterizations. It turns out that these rings can be characterized in terms of generalized pivotal monomials, thereby yielding
a theory similar to that of [17].
An erratum to this article is available at . 相似文献
5.
6.
Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined. 相似文献
7.
Ramamurthi proved that weak regularity is equivalent to regularity and biregularity for left Artinian rings. We observe this result under a generalized condition. For a ring R satisfying the ACC on right annihilators, we actually prove that if R is left weakly regular then R is biregular, and that R is left weakly regular if and only if R is a direct sum of a finite number of simple rings. Next we study maximality of strongly prime ideals, showing that a reduced ring R is weakly regular if and only if R is left weakly regular if and only if R is left weakly π-regular if and only if every strongly prime ideal of R is maximal. 相似文献
8.
In this paper, we consider a topological prime quasi-radical μ(R), which is the intersection of closed prime ideals in a topological ring R. Examples are given that show that μ(R) is different from those topological analogs of the prime radical that have been studied earlier. The topological prime quasi-radicals
of matrix rings and rings of polynomials are investigated.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 11–22, 2004. 相似文献
9.
The structure of rings all of whose ideals are prime is studied and several examples of such rings are constructed. 相似文献
10.
Let R be a ring, (S,≤) a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series. 相似文献
11.
M.A. Chebotar 《代数通讯》2013,41(6):2767-2777
We describe Lie isomorphisms of skew elements of prime rings of characteristic not 2 with involution of the first kind thus extending the corresponding result of Beidar, Martindale and Mikhalev to the case of characteristic 3. 相似文献
12.
L. J. Ratliff Jr. 《Annali di Matematica Pura ed Applicata》1982,130(1):39-58
Summary
The following topics are considered: saturated chains of prime ideals in quadratic (resp., simple, local, and level) integral extension domains of a local domain R; the heights of a prime ideal and its contraction in integral extension domains of R; and, the nonexistence of intermediate rings between R and finite integral extension domains of R that are minimal with spect to having certain properties.Research on this paper was supported in part by the National Science Foundation, Grant MCS 8001597. 相似文献
13.
Edward S. Letzter 《Israel Journal of Mathematics》2012,192(1):67-81
We study prime ideals in skew power series rings T:= R[[y; τ, δ]], for suitably conditioned complete right Noetherian rings R, automorphisms τ of R, and τ-derivations δ of R. Such rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern “Cutting Down” and “Lying Over.” In particular, assuming that τ extends to a compatible automorphsim of T, we prove: If I is an ideal of R, then there exists a τ-prime ideal P of T contracting to I if and only if I is a τ-δ-prime ideal of R. Consequently, under the more specialized assumption that δ = τ ? id (a basic feature of the Iwasawa-theoretic context), we can conclude: If I is an ideal of R, then there exists a prime ideal P of T contracting to I if and only if I is a τ-prime ideal of R. Our approach depends essentially on two key ingredients: First, the algebras considered are Zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings. 相似文献
14.
We extend some known results on radicals and prime ideals from polynomial rings and Laurent polynomial rings to Z-graded rings, i.e, rings graded by the additive group of integers. The main of them concerns the Brown–McCoy radical G and the radical S, which for a given ring A is defined as the intersection of prime ideals I of A such that A/I is a ring with a large center. The studies are related to some open problems on the radicals G and S of polynomial rings and situated in the context of Koethe’s problem. 相似文献
15.
Hull-kernel topology on the set ∑(R) of prime ideals of a ring R with unity and without nilpotent elements is discussed. The restriction of this topology to the set π(R) of minimal prime ideals of R has been investigated in detail. The compactness of π(R) has been characterized in several ways. An interesting characterization of Baer rings is given.A functorial correspondence between the category of rings having the property that every prime ideal contains a unique minimal prime ideal and their minimal spectra is established. 相似文献
16.
Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated
modules. The failure of this conjecture—by a 2003 counterexample due to Jorgensen and Şega—motivates the consideration of
the class of rings that do satisfy Auslander’s condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein.
Furthermore, the Auslander–Reiten Conjecture is proved for AC rings, and Auslander’s G-dimension is shown to be functorial
for AC rings that are commutative or have a dualizing complex. 相似文献
17.
We give criteria for the Cohen-Macaulay and Gorenstein property of Rees algebras of height 2 non-singular equimultiple prime
ideals in terms of explicite representations of the associated graded rings. As consequences, we show that in general, the
Cohen-Macaulay resp. Gorenstein property of such Rees algebras does not imply the Cohen-Macaulay resp. Gorenstein property
of the base ring and that these properties depend upon the characteristic.
Dedicated to the memrory of Professor Lê Van Thiêm
Professor Lê Van Thiêm was the first directorof Hanoi Institute of Mathematics 相似文献
18.
Francesc Planas-Vilanova 《Journal of Pure and Applied Algebra》2021,225(2):106494
Let R be a Noetherian ring. We prove that R has global dimension at most two if, and only if, every prime ideal of R is of linear type. Similarly, we show that R has global dimension at most three if, and only if, every prime ideal of R is syzygetic. As a consequence, we derive a characterization of these rings using the André-Quillen homology. 相似文献
19.
We study compact rings satisfying a pure algebraic condition, namely the periodicity of the group of units. Surprisingly,
this class of compact rings is related to the class of locally finite rings. We consider also more general classes or rings
with periodic group of rings: linearly compact rings. Properties of groups of units of compact rings were studied in [1-6].
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
20.
Byungchul Cha 《Journal of Number Theory》2010,130(4):1048-1055
We derive a formula for the density of positive integers satisfying a certain system of inequality, often referred as prime number races, in the case of the polynomial rings over finite fields. This is a function field analog of the work of Feuerverger and Martin, who established such formula in the number field case, building up on the fundamental work of Rubinstein and Sarnak. 相似文献