The structure of Johns rings

Faith, Carl; Menal, Pere

doi:10.1090/s0002-9939-1994-1231294-8

Cited by 19 publications

(4 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are three unresolved Faith conjectures on QF rings (see [10]). One of them is the Faith-Menal conjecture, which was raised by Faith and Menal in [3]. The conjecture says that every strongly right Johns ring is QF.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A NOTE ON ℵ₀-INJECTIVE RINGS

Shen

2011

J. Algebra Appl.

View full text Add to dashboard Cite

A ring R is called right ℵ 0 -injective if every right homomorphism from a countably generated right ideal of R to R R can be extended to a homomorphism from R R to R R . In this note, some characterizations of ℵ 0 -injective rings are given. It is proved that if R is semiperfect, then R is right ℵ 0injective if and only if every homomorphism from a countably generated small right ideal of R to R R can be extended to one from R R to R R . It is also shown that if R is right noetherian and left ℵ 0 -injective, then R is QF. This result can be looked as an approach to the Faith-Menal conjecture.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Later in [2], Faith and Menal gave a counter example to show that right Johns rings may not be right artinian. They characterized strongly right Johns rings as right noetherian and left FP-injective rings (see [3,Theorem 1.1]). So the Faith-Menal conjecture is equivalent to say that every right noetherian and left FP-injective ring is QF.…”

Section: Introductionmentioning

confidence: 99%

A NOTE ON ℵ₀-INJECTIVE RINGS

Shen

2011

J. Algebra Appl.

View full text Add to dashboard Cite

show abstract

“…As applications, we show that the Faith-Menal conjecture is true if R R is strongly finite dimensional. The Faith-Menal conjecture was raised by Faith and Menal in [3]. It says that every strongly right Johns ring is QF .…”

Section: Introductionmentioning

confidence: 99%

“…In [2], Faith and Menal gave a counter example to show that right Johns rings need not be right artinian. Later (see [3]) they defined strongly right Johns ring(the matrix ring M n×n (R) is right Johns for all n ≥ 1) and characterized such rings as right noetherian and left F P -injective rings. But they didn't know whether a strongly right Johns ring is QF .…”

Section: Introductionmentioning

confidence: 99%

On Strong Goldie Dimension

Shen

Chen

2007

Communications in Algebra

View full text Add to dashboard Cite

Let R be an associative ring with identity. A unital right Rmodule M is called strongly finite dimensional if Sup{G.dim(M/N ) | N ≤ M } < +∞. Properties of strongly finite dimensional modules are explored. It is also proved that: (1)If R is left F -injective and strongly right finite dimensional, then R is left finite dimensional. (2) If R is right Finjective, then R is right finite dimensional if and only if R is semilocal. Thus the Faith-Menal conjecture is true if R is strongly right finite dimensional. Some known results are obtained as corollaries.

show abstract

Annihilators and the CS-condition

Nicholson

Yousif

1998

Glasgow Math. J.

View full text Add to dashboard Cite

Abstract. It is proved that if every cyclic right /^-module is torsionless and R is a left CS-ring then R is semiperfect left continuous with soc(R R ) essential in R R. As a consequence every right cogenerator, left CS-ring R is shown to be right pseudo-Frobenius and left continuous, and an example is given to show that R need not be left selfinjective. It is also proved that if R is a left CS-ring and every cyclic right /^-module embeds in a free module, then R is quasi-Frobenius if and only if J(R) c Z(R R ). Introduction.Right CS-rings with certain cogenerating (annihilator) conditions were considered by Gomez-Pardo and Guil Asensio in [10] and [11]. For example they show that if R is a right CS-ring and every cyclic (finitely generated) right 7?-module embeds in a free module then R is right artinian (quasi-Frobenius). They also prove that if R is a right cogenerator right CS-ring then R is right pseudo-Frobenius.In this paper we consider these same classes of rings but with the left CS-condition rather than the right CS-condition. We show that if R is a left CS-ring and every cyclic right /^-module is torsionless, then R is a semiperfect left continuous ring with SOC(RR) C.

show abstract

The structure of Johns rings

Abstract: Abstract. In this paper we continue our study of right Johns rings, that is,

Cited by 19 publications

References 14 publications

A NOTE ON ℵ₀-INJECTIVE RINGS

A NOTE ON ℵ₀-INJECTIVE RINGS

On Strong Goldie Dimension

Annihilators and the CS-condition

Contact Info

Product

Resources

About

The structure of Johns rings

Abstract: Abstract. In this paper we continue our study of right Johns rings, that is,

Cited by 19 publications

References 14 publications

A NOTE ON ℵ0-INJECTIVE RINGS

A NOTE ON ℵ0-INJECTIVE RINGS

On Strong Goldie Dimension

Annihilators and the CS-condition

Contact Info

Product

Resources

About

A NOTE ON ℵ₀-INJECTIVE RINGS

A NOTE ON ℵ₀-INJECTIVE RINGS