THE FGF CONJECTURE AND THE SINGULAR IDEAL OF A RING

2013 ◽  
Vol 12 (07) ◽  
pp. 1350025 ◽  
Author(s):  
JOSÉ GÓMEZ-TORRECILLAS ◽  
PEDRO A. GUIL ASENSIO
Keyword(s):  
Left Ideal ◽  
Jacobson Radical ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Singular Ideal

We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its left singular ideal. We deduce that a left FGF is Quasi-Frobenius (QF) under this assumption. This clarifies the role played by the Jacobson radical and the singular left ideal in the FGF and CF conjectures. In Sec. 3 of the paper, we study the structure of left artinian left CF rings. We prove that they are left continuous and left CEP rings.

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.

Continuous Rings and Rings of Quotients

1978 ◽  
Vol 21 (3) ◽  
pp. 319-324 ◽  
Author(s):  
S. S. Page
Keyword(s):  
Jacobson Radical ◽  
Associative Ring ◽  
Natural Generalization ◽  
Injective Hull ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Rings Of Quotients

Throughout R will denote an associative ring with identity. Let Zℓ(R) be the left singular ideal of R. It is well known that Zℓ(R) = 0 if and only if the left maximal ring of quotients of R, Q(R), is Von Neumann regular. When Zℓ(R) = 0, q(R) is also a left self injective ring and is, in fact, the injective hull of R. A natural generalization of the notion of injective is the concept of left continuous as studied by Utumi [4]. One of the major obstacles to studying the relationships between Q(R) and R is a description of J(Q(R)), the Jacobson radical of Q(R). When a ring is left continuous, then its left singular ideal is its Jacobson radical. This facilitates the study of the cases when either Q(R) is continuous or R is continuous.

Stable Rings

1980 ◽  
Vol 23 (2) ◽  
pp. 173-178 ◽  
Author(s):  
S. S. Page
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Commutative Rings ◽  
Von Neumann ◽  
Goldie Dimension ◽  
Von Neumann Regular

Let R be an associative ring with identity. If R is von- Neumann regular of a left v-ring, then for each left ideal, I, we have I2 = I. In this note we study rings such that for each left ideal L there exists an integer n = n(L)>0 such that Ln = Ln+1. We call such rings stable rings. We completely describe the stable commutative rings. These descriptions give rise to concepts related to, but more general than, finite Goldie dimension and T-nilpotence, and a notion of power pure.

The Singular Congruence and the Maximal Quotient Semigroup

1972 ◽  
Vol 15 (2) ◽  
pp. 301-303 ◽  
Author(s):  
F. R. McMorris
Keyword(s):  
Quotient Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Singular Ideal ◽  
Quotient Semigroup

It is a well known result (see [4, p. 108]) that if R is a ring and Q(R) its maximal right quotient ring, then Q(R) is (von Neumann) regular if and only if every large right ideal of R is dense. This condition is equivalent to saying that the singular ideal of R is zero. In this note we show that the condition loses its magic in the theory of semigroups.

Modules Whose Endomorphism Rings Are (m, n)-Coherent

2019 ◽  
Vol 26 (02) ◽  
pp. 231-242
Author(s):  
Xiaoqiang Luo ◽  
Lixin Mao
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Coherent Ring ◽  
Left Annihilator

Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator [Formula: see text] is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of Mn if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.

scholarly journals J-Regular Rings with Injectivities

2013 ◽  
Vol 20 (02) ◽  
pp. 343-347 ◽  
Author(s):  
Liang Shen
Keyword(s):  
Jacobson Radical ◽  
Regular Ring ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Regular Rings

Let R be a J-regular ring, i.e., R/J(R) is a von Neumann regular ring, where J(R) is the Jacobson radical of R. It is proved: (i) For every n ≥ 1, R is right n-injective if and only if every homomorphism from an n-generated small right ideal of R to RR can be extended to one from RR to RR. (ii) R is right FP-injective if and only if R is right (J,R)-FP-injective. Some known results are improved.

Rings All of whose Factor Rings are Semi-Prime

1969 ◽  
Vol 12 (4) ◽  
pp. 417-426 ◽  
Author(s):  
R.C. Courter
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Title One ◽  
Regular Rings

We prove in this paper that fifteen classes of rings coincide with the class of rings named in the title. One of them is the class of rings R such that X2 = X for each R-ideal X: we shall refer to rings with this property (and thus to the rings of the title) as fully idempotent rings. The simple rings and the (von Neumann) regular rings are fully idempotent. Indeed, every finitely generated right or left ideal of a regular ring is generated by an idempotent [l, p. 42], so that X2 = X holds for every one-sided ideal X.

Regular Rings are Very Regular

1982 ◽  
Vol 25 (1) ◽  
pp. 118-118 ◽  
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Finitely Generated ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Regular Rings

The following problem arose in a conversation with Abraham Zaks: “Suppose R is an associative ring with identity such that every finitely generated left ideal is generated by idempotents. Is R von-Neumann regular?” In the literature the “s” in “idempotents” is missing, and is replaced by “an idempotent”. The answer is, “Yes!”

scholarly journals On (von Neumann) regular rings

1974 ◽  
Vol 19 (1) ◽  
pp. 89-91 ◽  
Author(s):  
R. Yue Chi Ming
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Associative Ring ◽  
Simple Module ◽  
Short Note ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Left Annihilator ◽  
Regular Rings

Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.

Additive decomposition of ideals

2018 ◽  
Vol 17 (05) ◽  
pp. 1850085 ◽  
Author(s):  
Ali Mohammad Karparvar ◽  
Babak Amini ◽  
Afshin Amini ◽  
Habib Sharif
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Polynomial Identity ◽  
Jacobson Radical ◽  
Affirmative Answer ◽  
Additive Decomposition ◽  
Von Neumann ◽  
Unital Ring ◽  
Von Neumann Regular ◽  
Special Case

In this paper, we investigate decomposition of (one-sided) ideals of a unital ring [Formula: see text] as a sum of two (one-sided) ideals, each being idempotent, nil, nilpotent, T-nilpotent, or a direct summand of [Formula: see text]. Among other characterizations, we prove that in a polynomial identity ring every (one-sided) ideal is a sum of a nil (one-sided) ideal and an idempotent (one-sided) ideal if and only if the Jacobson radical [Formula: see text] of [Formula: see text] is nil and [Formula: see text] is von Neumann regular. As a special case, these conditions for a commutative ring [Formula: see text] are equivalent to [Formula: see text] having zero Krull dimension. While assuming Köthe’s conjecture in several occasions to be true, we also raise a question, the affirmative answer to which leads to the truth of the conjecture.

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