Homological dimensions and regular rings

In this paper, we study the class of rings in which every pure ideal is projective. We refer to rings with this property as PIP-rings. Some properties and examples of PIP-rings are given. When R is a PIP-ring, some new homological dimensions for complexes are given. As applications, we give some new characterizations of von Neumann regular rings, F-rings and semisimple Artinian rings.

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RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

Journal of Algebra and Its Applications ◽

10.1142/s0219498809003527 ◽

2009 ◽

Vol 08 (05) ◽

pp. 601-615

Author(s):

JOHN D. LAGRANGE

Keyword(s):

Commutative Rings ◽

Von Neumann ◽

Von Neumann Regular Rings ◽

Von Neumann Regular ◽

Ring Of Quotients ◽

Rings Of Quotients ◽

Special Case ◽

Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

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On (Strongly) Gorenstein Von Neumann Regular Rings

Communications in Algebra ◽

10.1080/00927872.2010.501773 ◽

2011 ◽

Vol 39 (9) ◽

pp. 3242-3252 ◽

Cited By ~ 8

Author(s):

Najib Mahdou ◽

Mohammed Tamekkante ◽

Siamak Yassemi

Keyword(s):

Von Neumann ◽

Von Neumann Regular Rings ◽

Von Neumann Regular ◽

Regular Rings

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Homological dimensions of smooth crossed products

Annals of Functional Analysis ◽

10.1007/s43034-021-00125-w ◽

2021 ◽

Vol 12 (3) ◽

Author(s):

Petr Kosenko

Keyword(s):

Crossed Products ◽

Homological Dimensions

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Dominant and global dimension of blocks of quantised Schur algebras

Mathematische Zeitschrift ◽

10.1007/s00209-021-02792-w ◽

2021 ◽

Author(s):

Ming Fang ◽

Wei Hu ◽

Steffen Koenig

Keyword(s):

Hecke Algebras ◽

Symmetric Groups ◽

Global Dimension ◽

Group Algebras ◽

Dominant Dimension ◽

Schur Algebras ◽

Homological Dimensions ◽

Weyl Duality

AbstractGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

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On feckly polar rings

Journal of Algebra and Its Applications ◽

10.1142/s021949881950021x ◽

2019 ◽

Vol 18 (02) ◽

pp. 1950021

Author(s):

Tugce Pekacar Calci ◽

Huanyin Chen

Keyword(s):

Triangular Matrix ◽

Matrix Rings ◽

Structure Theorems ◽

Regular Rings

In this paper, we introduce a new notion which lies properly between strong [Formula: see text]-regularity and pseudopolarity. A ring [Formula: see text] is feckly polar if for any [Formula: see text] there exists [Formula: see text] such that [Formula: see text] Many structure theorems are proved. Further, we investigate feck polarity for triangular matrix and matrix rings. The relations among strongly [Formula: see text]-regular rings, pseudopolar rings and feckly polar rings are also obtained.

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