Q-Divisible Modules

1971 ◽  
Vol 14 (4) ◽  
pp. 491-494 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Quotient Ring ◽  
Torsion Class ◽  
Direct Sums ◽  
Factor Module ◽  
Left Quotient ◽  
Divisible Modules

Let R be a ring with 1 and let Q denote the maximal left quotient ring of R [6]. In a recent paper [12], Wei called a (left). R-module M divisible in case HomR (Q, N)≠0 for each nonzero factor module N of M. Modifying the terminology slightly we call such an R-module a Q-divisible R-module. As shown in [12], the class D of all Q-divisible modules is closed under factor modules, extensions, and direct sums and thus is a torsion class in the sense of Dickson [5].

Quotient Rings of a Class of Lattice-Ordered-Rings

1973 ◽  
Vol 25 (3) ◽  
pp. 627-645 ◽  
Author(s):  
Stuart A. Steinberg
Keyword(s):  
Identity Element ◽  
Quotient Ring ◽  
Ring Extension ◽  
Qf Ring ◽  
Left Quotient ◽  
Quotient Rings

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R(i) (d1q)+ Λ (d2q)- = 0, and(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.We remark that this characterization holds even when R does not have an identity element.

On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

1972 ◽  
Vol 24 (4) ◽  
pp. 703-712 ◽  
Author(s):  
A. G. Heinicke
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Torsion Class ◽  
Ring Of Quotients ◽  
Ore Condition ◽  
The Right ◽  
Necessary And Sufficient

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

scholarly journals Commutative rings whose quotients are Goldie

1975 ◽  
Vol 16 (1) ◽  
pp. 32-33 ◽  
Author(s):  
Victor P. Camillo
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
The Other ◽  
Direct Sums ◽  
Other Hand ◽  
Commutative Domain ◽  
Ascending Chain Condition

All rings considered here have units. A (non-commutative) ring is right Goldieif it has no infinite direct sums of right ideals and has the ascending chain condition on annihilator right ideals. A right ideal A is an annihilator if it is of the form {a ∈ R/xa = 0 for all x ∈ X}, where X is some subset of R. Naturally, any noetherian ring is Goldie, but so is any commutative domain, so that the converse is not true. On the other hand, since any quotient ring of a noetherian ring is noetherian, it is true that every quotient is Goldie. A reasonable question therefore is the following: must a ring, such that every quotient ring is Goldie, be noetherian? We prove the following theorem:Theorem. A commutative ring is noetherian if and only if every quotient is Goldie.

scholarly journals TTF-classes over perfect rings

1970 ◽  
Vol 11 (4) ◽  
pp. 499-503 ◽  
Author(s):  
J. S. Alin ◽  
E. P. Armendariz
Keyword(s):  
Torsion Class ◽  
Direct Sums

For a ring R with unit, let RM denote the category of unitary left R-modules. Following S.E. Dickson [3], a(non-empty) class of R-modules is a torsion class in rM if is closed under factors, extensions, and direct sums. If, in addition, is closed under submodules, then is said to be hereditary.

scholarly journals The classical left regular left quotient ring of a ring and its semisimplicity criteria

2017 ◽  
Vol 16 (06) ◽  
pp. 1750103
Author(s):  
V. V. Bavula
Keyword(s):  
Quotient Ring ◽  
Regular Elements ◽  
Left Regular ◽  
Left Quotient

Let [Formula: see text] be a ring, [Formula: see text] and [Formula: see text] be the set of regular and left regular elements of [Formula: see text] ([Formula: see text]). Goldie’s Theorem is a semisimplicity criterion for the classical left quotient ring [Formula: see text]. Semisimplicity criteria are given for the classical left regular left quotient ring [Formula: see text]. As a corollary, two new semisimplicity criteria for [Formula: see text] are obtained (in the spirit of Goldie).

A Lattice Isomorphism Theorem for Nonsingular Retractable Modules

1994 ◽  
Vol 37 (1) ◽  
pp. 140-144 ◽  
Author(s):  
Zhou Zhengping
Keyword(s):  
Quotient Ring ◽  
Lattice Isomorphism ◽  
Injective Hull ◽  
Isomorphism Theorem ◽  
Left Quotient

AbstractLet RM be a nonsingular module such that B = EndR(M) is left nonsingular and has as its maximal left quotient ring, where is the injective hull of RM. Then it is shown that there is a lattice isomorphism between the lattice C(M) of all complement submodules of RM and the lattice C(B) of all complement left ideals of B, and that RM is a CS module if and only if B is a left CS ring. In particular, this is the case if RM is nonsingular and retractable.

Torsion Theories Induced by Tilting Modules

1984 ◽  
Vol 36 (5) ◽  
pp. 899-913 ◽  
Author(s):  
Ibrahim Assem
Keyword(s):  
Torsion Theory ◽  
Torsion Class ◽  
Tilting Module ◽  
Tilting Modules ◽  
Direct Sums ◽  
Commutative Field ◽  
Finite Dimensional ◽  
Torsion Theories ◽  
Image Position ◽  
Torsion Free

Let k be a commutative field, and A a finite-dimensional k-algebra. By a module will always be meant a finitely generated right module. Following [8], we shall call a module TA a tilting module if (1) pdTA ≦ 1, (2) Ext1A(T, T) = 0 and (3) there is a short exact sequencewith T’ and T” direct sums of direct summands of T. Given a tilting module TA, the full subcategories andof the category modA of A -modules are respectively the torsion-free class and the torsion class of a torsion theory on modA[8]. The aim of the present paper is to find conditions on a torsion theory in order that it be induced by a tilting module.

Maximal Quotient Rings and S-Rings

1972 ◽  
Vol 24 (5) ◽  
pp. 835-850 ◽  
Author(s):  
E. P. Armendariz ◽  
Gary R. McDonald
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Quotient Ring ◽  
Injective Envelope ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Left Quotient ◽  
Von Neumann Regular ◽  
Singular Ideal ◽  
Quotient Rings

Throughout, we assume all rings are associative with identity and all modules are unitary. See [7] for undefined terms and [3] for all homological concepts.Let R be a ring, E(R) the injective envelope of RR, and H =HomR(E(R),E(R)). Then we obtain a bimodule RE(R)H. Let Q = HomH(E(R), E(R)). Q is called the maximal left quotient ring of R. Q has the property that if p, q ∈ Q, p ≠ 0, then there exists r ∈ R such that rp ≠ 0, rq ∈ R, i.e., Q is a ring of left quotients of R.A left ideal I of R is dense if for every x,y ∈ R,x ≠ 0, there exists r ∈ R such that rx ≠ 0, ry ∈ I. An alternate description of Q is Q = {x ∈ E(RR) : (R : x) is a dense left ideal of R{, where (R : x) = {r ∈ R : rx ∈ R}.The left singular ideal of R is Zl(R) = {r ∈ R : lR(r) is an essential left ideal of R}, where lR(r) = {x ∈ R : xr = 0}. If Zl(R) = (0), then Q is a left self-injective von Neumann regular ring [7, § 4.5]. Most of the previous work on maximal left quotient rings has been done in this case.

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