On a Theorem of Fuller

Baba, Yoshitomo; Oshiro, Kiyoichi

doi:10.1006/jabr.1993.1005

Cited by 16 publications

(5 citation statements)

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“…The statement of Proposition 2 in [3] includes the hypothesis that (eR, Rf ) is an i-pair. However this hypothesis is not used in the proof, so the result should be stated as follows:…”

mentioning

confidence: 99%

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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“…The statement of Proposition 2 in [3] includes the hypothesis that (eR, Rf ) is an i-pair. However this hypothesis is not used in the proof, so the result should be stated as follows:…”

mentioning

confidence: 99%

On perfect simple-injective rings

Nicholson¹,

Yousif²

1997

Proc. Amer. Math. Soc.

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“…If 0 T p P P, then rp rP J and so rp J because R is local. As Dp Rp is simple, (1) gives Dp lrp lJ socR R l V V È PX As P T 0Y this gives l V V 0 and dim D P 1X Finally, if w P r V V and 0 T p P P, then w p and p are in soc R R so that rw p J rpX As before, (1) gives Dw p lrw p lrp DpX Since V È P is direct, this implies that w 0Y whence r V V 0X (2)A(3). Using Lemma 1(4), soc R R r V V È P P l V V È P socR R .…”

Section: Question 2 Is There a Converse To Theorem 2?mentioning

confidence: 99%

“…If 0 T p P P and u P l V V, and if X p D 3 u p D is given by p d u p dY then is R-linear by Lemma 1 (8). By (1), cÁ is left multi-plication by c P R and so u p p cp P PX Thus u 0Y whence l V V 0X If 0 T p P P, then pR pD is simple so that lrp Rp by (1). Hence Lemma 1 (4) gives…”

Section: (7)mentioning

confidence: 99%

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A Look at the Faith Conjecture

Ara¹,

Nicholson²,

Yousif

2000

Glasgow Math. J.

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Throughout this paper all rings are associative with unity, and all modules are unital. If R is a ring we write J JR for the Jacobson radical of RX The socle of a module M is denoted by socMX Annihilators of a subset X R are written lX fa P R j aX 0g and rX fa P R j Xa 0gX We write N ess M (respectively N max M to indicate that N is an essential (maximal) submodule of MX The symbol D will always denote a division ring.Generalities. If S is any ring and S V S Y S W S and S P S are bimodules, a function

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“…Let R be a right or left artinian ring and let e be a primitive idempotent in R. Then eR R is injective if and only if there exists a primitive idempotent f in R such that S(eR) » = f R=fJ and S(Rf ) » = Re=Je, where S(X) and J mean the socle of X and the Jacobson radical of R, respectively. In Baba-Oshiro [2], this theorem is improved for a semiprimary ring with \ACC or DCC" for right annihilator ideals, where ACC and DCC mean the ascending chain condition and the descending chain condition, respectively. As the condition ACC or DCC for right annihilator ideals is equivalent to the condition ACC or DCC for left annihilator ideals, the replacement of \right or left artinian" with \semiprimary ring with ACC or DCC for annihilator right ideals" is quite natural.…”

Section: Introductionmentioning

confidence: 99%

REMARKS ON QF-2 RINGS, QF-3 RINGS AND HARADA RINGS

Iwase

2008

Ring Theory 2007

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On a Theorem of Fuller

Cited by 16 publications

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On perfect simple-injective rings

On perfect simple-injective rings

A Look at the Faith Conjecture

REMARKS ON QF-2 RINGS, QF-3 RINGS AND HARADA RINGS

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