Annihilators and the CS-condition

Nicholson, W. K.; Yousif, Mohamed

doi:10.1017/s0017089500032535

Cited by 17 publications

(6 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, we prove that a semiperfect ring R is right PF if and only if JðRÞ ZðR R Þ and R cogenerates every 2-generated right R-module. As every left CS right Kasch ring is semiperfect [6], this extends [14,Theorem 2.8] where it is proved that a left CS ring R with JðRÞ ZðR R Þ which cogenerates every 2-generated right R-module is right PF. In section 2 we study some relationships between right mininjective, right minsymmetric and left minannihilator rings (that is, rings for which every minimal left ideal is a left annihilator).…”

supporting

confidence: 57%

Semiperfect rings and Nakayama permutations

Khurana

2002

Glasgow Math. J.

View full text Add to dashboard Cite

Abstract. We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its basic idempotents. We also give a few necessary and sufficient conditions for a semiperfect ring R, which cogenerates every 2-generated right R-module, to be right pseudo-Frobenius.2000 Mathematics Subject Classification. 16L30, 16L60.0. Introduction. Throughout R is an associative ring with identity and modules are unitary. The right and left annihilators of subset X of a ring R are denoted by r R ðXÞ and l R ðXÞ respectively. We write J ¼ JðRÞ for the Jacobson radical of a ring R and SocðMÞ for the socle of a module M. Right and left singular ideals of a ring R will be denoted by ZðR R Þ and Zð R RÞ respectively. By N j j j M we shall mean that N is an essential submodule of a module M.A ring R is called right mininjective (right principally injective) if every R-homomorphism from a minimal (principal) right ideal of R into R is given by left multiplication by an element of R. Mininjective rings were introduced by Harada [7] who studied them in Artinian case. Recently Nicholson and Yousif [13] studied arbitrary mininjective rings. Principally injective rings have been studied in [3,12,16,17]. A ring R is called right Kasch if R contains a copy of each simple right R-module. An idempotent e of a ring R is called local if eRe is a local ring; equivalently if eJ is the unique maximal submodule of eR. Nakayama [10] called a left and right Artinian ring R with basic set of idempotents e 1 ; :::; e n quasi-Frobenius if there exists a permutation of f1; :::; ng such that SocðRe ðiÞ Þ ffi Re i =Je i and Socðe i RÞ ffi e ðiÞ R=e ðiÞ J:Let R be a semiperfect ring with basic set of idempotents e 1 ; :::; e n . In this paper, following Nicholson and Yousif [12], we call a permutation of f1; :::; ng a Nakayama permutation if there exists a set k 1 ; :::; k n of elements of R such that for each i(1) Rk i Re ðiÞ and k i R e i R; (2) Rk i ffi Re i =Je i and k i R ffi e ðiÞ R=e ðiÞ J.In particular, fk 1 R; :::; k n Rg and fRk 1 ; :::; Rk n g are complete irredundant sets of representatives of isomorphism classes of simple right and simple left R-modules respectively and so R is left and right Kasch.If a ring is right self injective and right cogenerator, it is called right pseudoFrobenius (PF). Extending some well known results on PF and quasi-Frobenius (QF) rings, Nicholson and Yousif proved that a right minfull ring (that is a semiperfect right mininjective ring R with SocðeRÞ 6 ¼ 0 for every local idempotent e [13])

show abstract

supporting

confidence: 57%

Semiperfect rings and Nakayama permutations

Khurana

2002

Glasgow Math. J.

View full text Add to dashboard Cite

show abstract

“…By using a method essentially due to Rada and Saorin, we obtain that a right CF ring is right artinian if and only if it is left perfect or semilocal with essential right socle. In particular, every right Johns and left CS-ring is QF (Nicholson and Yousif, 1998). Finally, we prove that a right noetherian left P-injective and left CS-ring is QF.…”

Section: Introductionmentioning

confidence: 76%

“…Nicholson and Yousif (1998) have shown that every right CF and right perfect ring is right artinian. Nicholson and Yousif (1998) have shown that every right CF and right perfect ring is right artinian.…”

Section: Proof For Any Two Right Idealsmentioning

confidence: 99%

See 1 more Smart Citation

On Artiness of Right CF Rings

Chen

2004

Communications in Algebra

View full text Add to dashboard Cite

show abstract

“…(1)⇔( 3) is clear by Lemma 2.4. ( 1)⇔( 2) can be obtained from [12,Lemma 2.1]. For the sake of completeness, we provide the proof.…”

Section: Resultsmentioning

confidence: 99%

On small dual rings

Shen¹

2014

J. Algebra Appl.

View full text Add to dashboard Cite

A ring R is called right (small) dual if every (small) right ideal of R is a right annihilator. Left (small) dual rings can be defined similarly. And a ring R is called (small) dual if R is left and right (small) dual. It is proved that R is a dual ring if and only if R is a semilocal and small dual ring. Several known results are generalized and properties of small dual rings are explored. As applications, some characterizations of QF rings are obtained through small dualities of rings.

show abstract

Annihilators and the CS-condition

Cited by 17 publications

References 17 publications

Semiperfect rings and Nakayama permutations

Semiperfect rings and Nakayama permutations

On Artiness of Right CF Rings

On small dual rings

Contact Info

Product

Resources

About