scholarly journals Lattice-ordered modules of quotients

1980 ◽  
Vol 30 (2) ◽  
pp. 243-251 ◽  
Author(s):  
Stuart A. Steinberg
Keyword(s):  
Jacobson Radical ◽  
Torsion Theory ◽  
Finitely Generated ◽  
Hereditary Torsion Theory ◽  
Ring Of Quotients

AbstractLet Q be the ring of quotients of the f-ring R with respect to a positive hereditary torsion theory and suppose Q is a right f-ring. It is shown that if the finitely-generated right ideals of R are principal, then Q is an f-ring. Also, if QR is injective, Q is an f-ring if and only if its Jacobson radical is convex. Moreover, a class of po-rings is introduced (which includes the classes of commutative po-rings and right convex f-rings) over which Q(M) is an f-module for each f-module M.

scholarly journals Torsion theories and coherent rings

1973 ◽  
Vol 8 (2) ◽  
pp. 233-239 ◽  
Author(s):  
J.M. Campbell
Keyword(s):  
Direct Product ◽  
Torsion Theory ◽  
Finitely Generated ◽  
Coherent Ring ◽  
Ring Of Quotients ◽  
Torsion Theories ◽  
Coherent Rings ◽  
Finitely Related

Chase has given several characterizations of a right coherent ring, among which are: every direct product of copies of the ring is left-flat; and every finitely generated submodule of a free right module is finitely related. We extend his results to obtain conditions for the ring of quotients of a ring with respect to a torsion theory to be coherent.

CHAIN CONDITIONS ON NON-SUMMANDS

2011 ◽  
Vol 10 (03) ◽  
pp. 475-489 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
A. ÇIĞDEM ÖZCAN ◽  
PATRICK F. SMITH
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Chain Conditions ◽  
Finite Dimensional

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes [Formula: see text], such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

scholarly journals Hereditary Torsion Theory Counter Equivalences

1996 ◽  
Vol 183 (1) ◽  
pp. 217-230 ◽  
Author(s):  
R.R. Colby ◽  
K.R. Fuller
Keyword(s):  
Torsion Theory ◽  
Hereditary Torsion Theory

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

scholarly journals Generating Adjoint Groups

2019 ◽  
Vol 62 (3) ◽  
pp. 733-738 ◽  
Author(s):  
Be'eri Greenfeld
Keyword(s):  
Open Problem ◽  
Jacobson Radical ◽  
The Other ◽  
Adjoint Group ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Other Hand ◽  
Nil Ring

AbstractWe prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated. We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated and, on the other hand, construct a finitely generated, infinite-dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.

Coherence relative to an hereditary torsion theory

1982 ◽  
Vol 10 (7) ◽  
pp. 719-739 ◽  
Author(s):  
Marsha Finkel Jones
Keyword(s):  
Torsion Theory ◽  
Hereditary Torsion Theory

scholarly journals Rings all of whose torsion quasi-injective modules are injective

1984 ◽  
Vol 25 (2) ◽  
pp. 219-227 ◽  
Author(s):  
J. Ahsan ◽  
E. Enochs
Keyword(s):  
Jacobson Radical ◽  
Identity Element ◽  
Torsion Theory ◽  
Injective Hull ◽  
Torsion Class ◽  
Injective Modules ◽  
Torsion Theories ◽  
Torsion Free

Throughout this paper it is assumed that rings are associative, have the identity element, and all modules are left unital. R will denote a ring with identity, R-Mod the category of left R-modules, and for each left R-module M, E(M) (resp. J(M)) will represent the injective hull (resp. Jacobson radical) of M. Also, for a module M, A ⊆' M will mean that A is an essential submodule of M, and Z(M) denotes the singular submodule of M. M is called singular if Z(M) = M, and it is called non-singular in case Z(M) = 0. For fundamental definitions and results related to torsion theories, we refer to [12] and [14]. In this paper we shall deal mainly with Goldie torsion theory. Recall that a pair (G, F) of classes of left R-modules is known as Goldie torsion theory if G is the smallest torsion class containing all modules B/A, where A ⊆' B, and the torsion free class F is precisely the class of non-singular modules.

scholarly journals On group graded rings satisfying polynomial identities

1995 ◽  
Vol 37 (2) ◽  
pp. 205-210 ◽  
Author(s):  
A. V. Kelarev ◽  
J. Okniński
Keyword(s):  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Normal Band ◽  
Ring Theory ◽  
Semigroup Ring ◽  
Semigroup Algebras ◽  
Graded Rings ◽  
Finitely Generated ◽  
Polynomial Identities ◽  
Locally Finite

A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [10], [18]). Several interesting results of this sort have appeared in the literature recently. In particular, it was proved in [1] that the Jacobson radical of every finitely generated PI-ring is nilpotent. For every commutative semigroup ring RS, it was shown in [11] that if J(R) is nil then J(RS) is nil. This result was generalized to all semigroup algebras satisfying polynomial identities in [15] (see [16, Chapter 21]). Further, it was proved in [12] that, for every normal band B, if J(R) is nilpotent, then J(RB) is nilpotent. A similar result for special band-graded rings was established in [13, Section 6]. Analogous theorems concerning nilpotency and local nilpotency were proved in [2] for rings graded by finite and locally finite semigroups.

scholarly journals Contracted primes of the complete ring of quotients

1988 ◽  
Vol 38 (3) ◽  
pp. 373-375
Author(s):  
Frederick W. Call
Keyword(s):  
Commutative Ring ◽  
Compact Set ◽  
Finitely Generated ◽  
Dense Ideal ◽  
Complete Ring ◽  
Ring Of Quotients

The generic closure of the set of primes contracted from the complete ring of quotients of a reduced commutative ring is shown to be just the set of those primes not containing a finitely generated dense ideal. It is also the smallest generically closed, quasi-compact set containing the minimal primes.

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