Rings in Which Every Complement Right Ideal Is a Direct Summand

Chatters, A. W.; Hajarnavis, C. R.

doi:10.1093/qmath/28.1.61

Cited by 70 publications

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“…A ring is right (quasicontinuous) CS if it is (quasi-continuous) CS as a right iϋ-module [6]. Right self-injective rings and products of right Ore domains are right quasi-continuous [13 & 19].…”

Section: Theorem 9 Let R Be a (Quasi-)baer Ring Which Is Neither Redmentioning

confidence: 99%

Baer rings and quasicontinuous rings have a MDSN

Birkenmeier¹

1981

Pacific J. Math.

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Section: Theorem 9 Let R Be a (Quasi-)baer Ring Which Is Neither Redmentioning

confidence: 99%

Baer rings and quasicontinuous rings have a MDSN

Birkenmeier¹

1981

Pacific J. Math.

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“…This property, originated by Chatters and Hajarnavis in [21], ensures a rich structure theory for these classes. Although every module has an injective hull, it is usually hard to compute.…”

mentioning

confidence: 88%

Modules With Fi-Extending Hulls

2009

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Abstract. It is shown that every finitely generated projective module P R over a semiprime ring R has the smallest FI-extending essential module extension H FI (P R ) (called the absolute FI-extending hull of P R ) in a fixed injective hull of P R . This module hull is explicitly described. It is proved that Q FI (End(P R )) ∼ = End(H FI (P R )), where Q FI (End(P R )) is the smallest right FI-extending right ring of quotients of End(P R ) (in a fixed maximal right ring of quotients of End(P R )). Moreover, we show that a finitely generated projective module P R over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(P R ) is a quasi-Baer ring.

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“…Recall that a ring R is called right CS when every complement right ideal is a direct summand of R R or, equivalently, every right ideal is essential in a direct summand of R R (cf. [3]). CS rings and CS modules (defined in a similar way and called also extending modules) have been object of intensive study in recent years (cf.…”

Section: Corollary 32 Let R Be a Ring Such That E(r R ) Is A Projecmentioning

confidence: 99%

Essential embedding of cyclic modules in projectives

Pardo

Asensio

1997

Trans. Amer. Math. Soc.

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Abstract. Let R be a ring and E = E(R R ) its injective envelope. We show that if every simple right R-module embeds in R R and every cyclic submodule of E R is essentially embeddable in a projective module, then R R has finite essential socle. As a consequence, we prove that if each finitely generated right R-module is essentially embeddable in a projective module, then R is a quasiFrobenius ring. We also obtain several other applications and, among them: a) we answer affirmatively a question of Al-Huzali, Jain, and López-Permouth, by showing that a right CEP ring (i.e., a ring R such that every cyclic right module is essentially embeddable in a projective module) is always right artinian; b) we prove that if R is right FGF (i.e., any finitely generated right R-module embeds in a free module) and right CS, then R is quasi-Frobenius.

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Rings in Which Every Complement Right Ideal Is a Direct Summand

Cited by 70 publications

References 0 publications

Baer rings and quasicontinuous rings have a MDSN

Baer rings and quasicontinuous rings have a MDSN

Modules With Fi-Extending Hulls

Essential embedding of cyclic modules in projectives

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