Classical Artinian Rings and Related Topics

Baba, Yoshitomo; Oshiro, Kiyoichi

doi:10.1142/7451

Cited by 41 publications

(15 citation statements)

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“…Further, this collection of idempotents is unique up to conjugation by a unit. For more on general theory of serial rings the reader is referred to [38], or to more recent [3]. We will downsize to a more restrictive setting.…”

Section: Basicsmentioning

confidence: 99%

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When the Group Ring of a Finite Simple Group is Serial

2019

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Section: Basicsmentioning

confidence: 99%

“…For instance, A 5 with p = 3 occurs twice in this list: first as PSL 2 (4) in 2), and then as PSL 2 (5) in 3).…”

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confidence: 99%

When the Group Ring of a Finite Simple Group is Serial

2019

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“…Hence T p (G) is divisible, and so injective. Now it is straightforward to check that G is A ring is called a right Harada ring if it is a perfect ring over which every nonsmall right module contains a nonzero injective submodule (see [4]). This is equivalent to the condition that any injective right module is supplemented and supplement submodules are injective (see [4, Proof.…”

Section: Proposition 25 Every Nonzero Cyclic Right R-module Is a Timentioning

confidence: 99%

Rings and modules characterized by opposites of injectivity

Alizade

Büyükaşık

2014

Journal of Algebra

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In a recent paper, Aydoǧdu and López-Permouth have defined a module M to be N -subinjective if every homomorphism N → M extends to some E(N ) → M , where E(N ) is the injective hull of N . Clearly, every module is subinjective relative to any injective module. Their work raises the following question: What is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules, namely injectives? We show, using a dual opposite injectivity condition, that such a ring R is isomorphic to the direct product of a semisimple Artinian ring and an indecomposable ring which is (i) a hereditary Artinian serial ring with J 2 = 0; or (ii) a QF-ring isomorphic to a matrix ring over a local ring. Each case is viable and, conversely, (i) is sufficient for the said property, and a partial converse is proved for a ring satisfying (ii). Using the above mentioned classification, it is also shown that such rings coincide with the fully saturated rings of Trlifaj except, possibly, when von Neumann regularity is assumed. Furthermore, rings and abelian groups which satisfy these opposite injectivity conditions are characterized.

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“…In [4,9], a construction of local QF rings with Jacobson radical cubed zero is studied. In the former article, which is included in Chapter 10 of the book [2], several new examples of local graded QF rings with Jacobson radical cubed zero are given, and in the later book, rings with Jacobson radical cubed zero are discussed in a general theory of QF rings. In [5][6][7] we also studied the classification of local QF algebras over a field.…”

Section: Introductionmentioning

confidence: 99%