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.

A Generalization of the Concept of a Ring of Quotients

1971 ◽  
Vol 14 (4) ◽  
pp. 517-529 ◽  
Author(s):  
John K. Luedeman
Keyword(s):  
Left Ideal ◽  
Quotient Ring ◽  
Original Method ◽  
Duke Math ◽  
Injective Hull ◽  
Direct Products ◽  
Left Module ◽  
Matrix Rings ◽  
Ring Of Quotients ◽  
Quotient Rings

AbstractSanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

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.

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).

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.

scholarly journals New criteria for a ring to have a semisimple left quotient ring

2015 ◽  
Vol 14 (06) ◽  
pp. 1550090 ◽  
Author(s):  
V. V. Bavula
Keyword(s):  
Quotient Ring ◽  
Artinian Ring ◽  
Ring Theory ◽  
Arbitrary Ring ◽  
The Third ◽  
Left Quotient ◽  
Finite Set ◽  
New Ideas ◽  
New Criteria ◽  
Prime Case

Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of the paper is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring R the set ℳ of maximal left denominator sets of R is a non-empty set [V. V. Bavula, The largest left quotient ring of a ring, preprint (2011), arXiv:math.RA:1101.5107]: Theorem (The First Criterion). A ring R has a semisimple left quotient ring Q iff ℳ is a finite set, ⋂S∈ℳ ass (S) = 0 and, for each S ∈ ℳ, the ring S-1R is a simple left Artinian ring. In this case, Q ≃ ∏S∈ℳ S-1R. The Second Criterion is given via the minimal primes of R and goes further than the First one in the sense that it describes explicitly the maximal left denominator sets S via the minimal primes of R. The Third Criterion is close to Goldie's Criterion but it is easier to check in applications (basically, it reduces Goldie's Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets.

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