Rings of quotients of perfect rings

Storrer, Hans H.

doi:10.1007/bf01110089

Cited by 17 publications

(10 citation statements)

References 13 publications

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“…So sSe is also colocal by [20,Proposition 4.4 (b)]. (We note that [20,Proposition 4.41 is given for a maximal quotient ring. But the results hold for a left quotient ring by its proof.)…”

Section: Application Of Theorem Bmentioning

confidence: 94%

Some classes of qf-3 rings

Baba¹

2000

Communications in Algebra

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“…So sSe is also colocal by [20,Proposition 4.4 (b)]. (We note that [20,Proposition 4.41 is given for a maximal quotient ring. But the results hold for a left quotient ring by its proof.)…”

Section: Application Of Theorem Bmentioning

confidence: 94%

Some classes of qf-3 rings

Baba¹

2000

Communications in Algebra

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“…97]. For example, if P is right perfect, then P has a minimal dense left ideal, and this case has been studied by Storrer [17]. In particular, Storrer obtained the following result.…”

Section: Then (I + P(r))/p(r)mentioning

confidence: 99%

“…R is a left Artinian right 5-ring which is not right Artinian.Example. Let F be a field, and let is a left and right Artinian ring which is a right, but not a left, 5-ring[17, Example 7.2].…”

mentioning

confidence: 99%

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Maximal Quotient Rings and S-Rings

Armendariz¹,

McDonald²

1972

Can. j. math.

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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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“…The following is an example of ring R for which D^D 0 . Let K be any field and let R consist of all 3 x 3 matrices over K of the form As noted in [10], R=Q; moreover R is left Artinian and the right ideal A of all matrices of the form has zero left annihilator. By [2,Theorem 6.3], R has a simple left-i?-module S which is not a factor of an injective i?-module, hence S $ D 0 .…”

Section: Corollary If R Is Any Integral Domain Then Every Torsion-fmentioning

confidence: 99%

Q-Divisible Modules

Armendariz

1971

Can. math. bull.

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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].

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Rings of quotients of perfect rings

Cited by 17 publications

References 13 publications

Some classes of qf-3 rings

Some classes of qf-3 rings

Maximal Quotient Rings and S-Rings

Q-Divisible Modules

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