Rings with the principal extension property

Rutter, Edgar

doi:10.1080/00927877508822043

Cited by 37 publications

(12 citation statements)

References 8 publications

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“…It was proved in [7,Theorem] that, if R is right P-injective and has ACC on annihilator right ideals, then R is left artinian. But we do not know if the ring S in Corollary 1.13(2) is left artinian.…”

Section: Annihilator Conditions 2315mentioning

confidence: 99%

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Modules With Annihilator Conditions

Ding¹,

Yousif

Zhou

2002

Communications in Algebra

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Let S and R be rings. The objective is to study the bimodule S N R satisfying the annihilator conditions l N ðr R ðxÞÞ ¼ Sx for all x 2 N. This approach will clearly show how the ring R or the module N R is connected to the properties of the ring S through the annhilator condition. Specializing to the particular bimodule R R R or End ðNÞN R , we obtain some new results and known results as corollaries.

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Section: Annihilator Conditions 2315mentioning

confidence: 99%

“…For the detailed study of P-injective rings, we refer to [1][2][3][4][5][6][7][8] . Let S and R be rings.…”

Section: Introductionmentioning

confidence: 99%

Modules With Annihilator Conditions

Ding¹,

Yousif

Zhou

2002

Communications in Algebra

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“…Note that every homomorphism from a simple left R -module M to R is of the form The following example shows that the hypothesis "left min-CS and left C2" in Proposition 3.6 is not superfluous. [16,19] Proof It suffices to prove the last statement. Let M be a simple left R -module with injective envelope …”

Section: Thus R Is a Commutative Local Ring With Unique Maximal (Leftmentioning

confidence: 99%

On the existence of nonzero injective covers and projective envelopes of modules

Zhang¹,

Xian-mei²

2013

Turk J Math

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In general, the injective cover (projective envelope) of a simple module can be zero. A ring R is called a weakly left V-ring (strongly left Kasch ring) if every simple left R -module has a nonzero injective cover (projective envelope). It is proven that every nonzero left R -module has a nonzero injective cover if and only if R is a left Artinian weakly left V-ring. Dually, every nonzero left R -module has a nonzero projective envelope if and only if R is a left perfect right coherent strongly left Kasch ring. Some related rings and examples are considered.

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“…Recall that a ring R is left principally injective (fP-injective) if every principal right ideal is a right annihilator, and the ring R is left generalized principally injective (GP'-injective) if, for any 0 ^ a e R, there exists n > 0 such that a" ^ 0 and a"R is a right annihilator. The detailed discussion of left P-injective and left GP-injective rings can be found in [3,7,12,15,16,17,22,23,24,26]. Clearly, every left AP-injective ring is left P-injective and every left AGP-injective ring is left GP-injective.…”

Section: Introductionmentioning

confidence: 99%