Principally Injective Rings

Nicholson, W. K.; Yousif, Mohamed

doi:10.1006/jabr.1995.1117

Cited by 128 publications

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“…In [15] R is called a right GPF-ring (generalized pseudo-Frobenius ring) if R is semiperfect, right P-injective and SOC(RR) c e " R R . Here R is called right P-injective (principally injective) if every /?-homomorphism from a principal right ideal of R into R is given by left multiplication.…”

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confidence: 99%

Annihilators and the CS-condition

Nicholson

Yousif

1998

Glasgow Math. J.

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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.

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Annihilators and the CS-condition

Nicholson

Yousif

1998

Glasgow Math. J.

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“…In [14], a ring R is called a right generalized pseudo-Frobenius ring (a right GPF-ring) if R is semiperfect, right principally injective and has an essential right socle.…”

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On perfect simple-injective rings

Nicholson¹,

Yousif²

1997

Proc. Amer. Math. Soc.

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Abstract. Harada calls a ring R right simple-injective if every R-homomorphism with simple image from a right ideal of R to R is given by left multiplication by an element of R. In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if R is left perfect and right simple-injective, then R is quasi-Frobenius if and only if the second socle of R is countably generated as a left R-module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings. Throughout this paper all rings R considered are associative with unity and all modules are unitary R-modules. We write M R to indicate a right R-module. The socle of a module is denoted by soc(M ). We write N ⊆ M (N ⊆ ess M ) to mean that N is a submodule (essential) of M . For any subset X of R, l(X) and r(X) denote, respectively, the left and right annihilators of X in R. A ring R is called quasi-Frobenius ifA ring R is called right Kasch if every simple right R-module is isomorphic to a minimal right ideal of R. The ring R is called right pseudo-Frobenius (a right PF-ring) if R R is an injective cogenerator in mod-R; equivalently if R is semiperfect, right self-injective and has an essential right socle.A ring R is called right principally injective if every R-morphism from a principal right ideal of R into R is given by left multiplication. In [14], a ring R is called a right generalized pseudo-Frobenius ring (a right GPF-ring) if R is semiperfect, right principally injective and has an essential right socle.We write J = J(R) for the Jacobson radical of the ring R. Following Fuller [10], if R is semiperfect with a basic set E of primitive idempotents, and if e, f ∈ E, we say that the pair (eR, Rf ) is an i-pair if soc(eR) ∼ = f R/f J and soc(Rf ) ∼ = Re/Je.

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“…Thus R is right Artinian by (3) and Corollary 2.6, and so R has A CC on left annihilators. By hypothesis, M 2 (R) is left P-injective, and so R is left 2-injective by [13,Theorem 4.2]. Thus R is QF by Lemma 2.11.…”

Section: A Ring R Is Called Right 2-gfmentioning

confidence: 82%

“…. , n. Since R is left P-injective, there exists an epimorphism <j>i : a t R -> niiR by [13 (2) => (1). Let R/A be a cyclic right /?-module, where A is a right ideal of R. Then R/A is torsionless (for A = r(l(A))) and finitely cogenerated because R is right Artinian.…”

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On Noetherian rings with essential socle

Chen¹,

Ding²,

Yousif³

2004

J. Aust. Math. Soc.

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It is shown that if R is a right Noetherian ring whose right socle is essential as a right ideal and is contained in the left socle, then R is right Artinian. This result may be viewed as a one-sided version of a result of Ginn and Moss on two-sided Noetherian rings with essential socle. This also extends the work of Nicholson and Yousif where the same result is obtained under a stronger hypothesis. We use our work to obtain partial positive answers to some open questions on right CF, right FGF and right Johns rings.2000 Mathematics subject classification: primary 16A33; secondary 16A35.

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Principally Injective Rings

Cited by 128 publications

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Annihilators and the CS-condition

Annihilators and the CS-condition

On perfect simple-injective rings

On Noetherian rings with essential socle

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