共查询到20条相似文献,搜索用时 15 毫秒
1.
《代数通讯》2013,41(9):3091-3119
ABSTRACT A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property. 相似文献
2.
J. E. van den Berg 《Acta Mathematica Hungarica》2000,87(1-2):153-172
This paper investigates closure properties possessed by certain classes of finite subdirect products of prime rings. If ℳ
is a special class of prime rings then the class ℳ∞ of all finite subdirect products of rings in ℳ is shown to be weakly special. A ring S is said to be a right tight extension [resp. tight extension] of a subring R if every nonzero right ideal [resp. right ideal and left ideal] of S meets R nontrivially. Every hereditary class of semiprime rings closed under tight extensions is weakly special. Each of the following
conditions imposed on a semiprime ring yields a hereditary class closed under right tight extensions: ACC on right annihilators;
finite right Goldie dimension; right Goldie. The class of all finite subdirect products of uniformly strongly prime rings
is shown to be closed under tight extensions, answering a published question.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
3.
Mabrouk Ben Nasr 《Monatshefte für Mathematik》2009,58(2):97-102
An extension of integral domains R í S{R\subseteq S} is said to have the “finite length of intermediate chains of domains” property (for short FICP) if each chain of intermediate
rings between R and S is finite. The main purpose of this paper is to characterize when R í S{R\subseteq S} has FICP in case R
* (the integral closure of R in S) is a finite dimensional semilocal domain. This generalizes a theorem due to Gilmer, in which S is the quotient field of R. Examples illustrating the sharpness and the limits of our results are settled. 相似文献
4.
V. V. Bavula 《代数通讯》2017,45(9):3798-3815
A new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S?1R of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals. 相似文献
5.
Ahmed Ayache 《Ricerche di matematica》2011,60(2):193-201
Let R ì S{R\subset S} be an extension of integral domains and let [R, S] be the set of intermediate rings between R and S ordered by inclusion. If (R, S) is normal pair and [R, S] is finite, we do prove that there exists a semi-local Prüfer ring T with quotient field K such that [R,S] @ [T,K]{[R,S]\cong \lbrack T,K]} (as a partially ordered set). Consequently, any problem relative to the finiteness conditions in [R, S] can be investigated in the particular case where R is a semi-local Prüfer ring with quotient field S. 相似文献
6.
Mabrouk Ben Nasr 《Monatshefte für Mathematik》2009,158(1):97-102
An extension of integral domains is said to have the “finite length of intermediate chains of domains” property (for short FICP) if each chain of intermediate
rings between R and S is finite. The main purpose of this paper is to characterize when has FICP in case R
* (the integral closure of R in S) is a finite dimensional semilocal domain. This generalizes a theorem due to Gilmer, in which S is the quotient field of R. Examples illustrating the sharpness and the limits of our results are settled.
相似文献
7.
《代数通讯》2013,41(12):6149-6159
Abstract A commutative ring R is said to satisfy property (P) if every finitely generated proper ideal of R admits a non-zero annihilator. In this paper we give some necessary and sufficient conditions that a ring satisfies property (P). In particular, we characterize coherent rings, noetherian rings and Π-coherent rings with property (P). 相似文献
8.
A right module M over a ring R is said to be retractable if Hom R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? R RG is a retractable RG-module if and only if M R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings. 相似文献
9.
Given two rings R ? S, S is said to be a minimal ring extension of R, if R is a maximal subring of S. In this article, we study minimal extensions of an arbitrary ring R, with particular focus on those possessing nonzero ideals that intersect R trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs &; Shapiro, and Ferrand &; Olivier, on commutative minimal extensions. 相似文献
10.
Sefi Ladkani 《Algebras and Representation Theory》2011,14(1):57-74
A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and
R
M
S
is an S-R-bimodule. In the main theorem of this paper we show that if T
S
is a tilting S-module, then under certain homological conditions on the S-module M
S
, one can extend T
S
to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and T
S
, and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and M
S
is finitely generated. In this case, (S′,R,M′) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, M
S
has a finite projective resolution and Ext
S
n
(M
S
, S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, Hom
S
(M, S)). 相似文献
11.
G. L. Cherlin 《Israel Journal of Mathematics》1976,25(1-2):87-96
SCR=the class of commutative rings without nilpotent elements.Theorem,R is an amalgamation base for SCR iff rad (I=Ann2(I) forI ⊆R finitely generated.
Supplement. IfR ε SCR thenR is contained in an amalgamation basis for SCR having no new idempotent elements.
CR=the class of commutative rings.
Theorem.R is an amalgamation base for CR iffR is a pureR-submodule of any commutative ring extendingR. 相似文献
12.
MP-injective rings and MGP-injective rings 总被引:1,自引:0,他引:1
Zhanmin Zhu 《印度理论与应用数学杂志》2010,41(5):627-645
A ring R is said to be right MP-injective if every monomorphism from a principal right ideal to R extends to an endomorphism of R. A ring R is said to be right MGP-injective if, for any 0 ≠ a ∈ R, there exists a positive integer n such that a
n
≠ 0 and every monomorphism from a
n
R to R extends to R. We shall study characterizations and properties of these two classes of rings. Some interesting results on these rings are
obtained. In particular, conditions under which right MGP-injective rings are semisimple artinian rings, von Neumann regular
rings, and QF-rings are given. 相似文献
13.
Peter Symonds 《Advances in Mathematics》2007,208(1):408-421
Given a polynomial ring R over a field k and a finite group G, we consider a finitely generated graded RG-module S. We regard S as a kG-module and show that various conditions on S are equivalent, such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of [D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: A finiteness theorem, preprint, http://www.ma.umist.ac.uk/pas/preprints]. 相似文献
14.
A widely used result of Wedderburn and Artin states that “every left ideal of a ring R is a direct summand of R if and only if R has a unique decomposition as a finite direct product of matrix rings over division rings.” Motivated by this, we call a module M virtually semisimple if every submodule of M is isomorphic to a direct summand of M and M is called completely virtually semisimple if every submodule of M is virtually semisimple. We show that the left R-module R is completely virtually semisimple if and only if R has a unique decomposition as a finite direct product of matrix rings over principal left ideal domains. This shows that R is completely virtually semisimple on both sides if and only if every finitely generated (left and right) R-module is a direct sum of a singular module and a projective virtually semisimple module. The Wedderburn-Artin theorem follows as a corollary from our result. 相似文献
15.
Leonard Dăuş 《Applied Categorical Structures》2011,19(6):859-863
We use the concept of a regular object with respect to another object in an arbitrary category, in order to obtain the transfer
of regularity in the sense of Zelmanowitz between the categories R −mod and S −mod, when S is an excellent extension of the ring R. Consequently, if S is an excellent extension of the ring R, then S is von Neumann regular ring if and only if R is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical
result: the von Neumann regular property of a ring is Morita invariant. 相似文献
16.
V.V. Bavula 《Journal of Pure and Applied Algebra》2019,223(3):998-1013
A new class of rings, the class of left localizable rings, is introduced. A ring R is left localizable if each nonzero element of R is invertible in some left localization of the ring R. Explicit criteria are given for a ring to be a left localizable ring provided the ring has only finitely many maximal left denominator sets (e.g., this is the case if a ring has a left Noetherian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets is a left localizable ring iff its left quotient ring is a direct product of finitely many division rings. A characterization is given of the class of rings that are finite direct product of left localization maximal rings. 相似文献
17.
Let R be a ring. An R-module M is finitely annihilated if the annihilator of M is the annihilator of a finite subset of M. It is proved that if R has right socle S then the ring R/S is right Artinian if and only if every singular right R-module is finitely annihilated. Moreover, a right Noetherian ring R is right Artinian if and only if every uniform right R-module is finitely annihilated. In addition, a (right and left) Noetherian ring is (right and left) Artinian if and only
if every injective right R-module is finitely annihilated.
This paper will form part of the Ph.D. thesis at the University of Glasgow of the second author. He would like to thank the
EPSRC for their financial support 相似文献
18.
PP-Rings of Generalized Power Series 总被引:6,自引:0,他引:6
Abstract
As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let
R be a commutative ring, and (S, ≤) a strictly totally ordered monoid. We prove that (1) the ring [[R
S,≤]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B(R) (the set of all idempotents of R) has a least upper bound in B(R) and (2) if (S, ≤) also satisfies the condition that 0 ≤s for any s∈S, then the ring [[R
S,≤
]] is weakly PP if and only if R is weakly PP.
Research supported by National Natural Science Foundation of China, 19501007, and Natural Science Foundation of Gansu, ZQ-96-01 相似文献
19.
A subset M of a universal algebra A is called finitely separated in A if, for any element x∈A/M, there exists an homomorphism
ϕ of A into a finite algebra, for which ϕ(x)∋ϕ(M). A ring is said to be S-separable (R-separable) if its subrings (resp.,
right ideals) are all finitely separated in it. We give equational (in the language of identities) and indicator (in the language
of “prohibited” rings) characterizations of varieties consisting of S-separable (R-separable) rings. Moreover, varieties are
described in which, not all, but finitely generated rings only share the properties mentioned.
Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 201–227, march–April, 1999. 相似文献
20.
We carry out an extensive study of modules M R with the property that M/f(M) is singular for all injective endomorphisms f of M. Such modules called “quasi co-Hopfian”, generalize co-Hopfian modules. It is shown that a ring R is semisimple if and only if every quasi co-Hopfian R-module is co-Hopfian. Every module contains a unique largest fully invariant quasi co-Hopfian submodule. This submodule is determined for some modules including the semisimple ones. Over right nonsingular rings several equivalent conditions to being quasi co-Hopfian are given. Modules with all submodules quasi co-Hopfian are called “completely quasi co-Hopfian” (cqcH). Over right nonsingular rings and over certain right Noetherian rings, it is proved that every finite reduced rank module is cqcH. For a right nonsingular ring which is right semi-Artinian (resp. right FBN) the class of cqcH modules is the same as the class of finite reduced rank modules if and only if there are only finitely many isomorphism classes of nonsingular R-modules which are simple (resp. indecomposable injective). 相似文献