On characterizations of w-coherent rings II

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

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Gröbner-coherent rings and modules

Journal of Commutative Algebra ◽

10.1216/jca.2020.12.107 ◽

2020 ◽

Vol 12 (1) ◽

pp. 107-114

Author(s):

Rohit Nagpal ◽

Andrew Snowden

Keyword(s):

Coherent Rings

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Special classes of generalized coherent rings

Acta Mathematica Academiae Scientiarum Hungaricae ◽

10.1007/bf01895232 ◽

1982 ◽

Vol 39 (1-3) ◽

pp. 185-193 ◽

Cited By ~ 3

Author(s):

F. T. Hannick

Keyword(s):

Coherent Rings ◽

Special Classes

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Relative homological groups over coherent rings

Quaestiones Mathematicae ◽

10.2989/16073606.2012.725273 ◽

2012 ◽

Vol 35 (3) ◽

pp. 331-352 ◽

Cited By ~ 2

Author(s):

Lixin Mao

Keyword(s):

Coherent Rings

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Relative FP-gr-injective and gr-flat modules

International Journal of Algebra and Computation ◽

10.1142/s021819671850042x ◽

2018 ◽

Vol 28 (06) ◽

pp. 959-977 ◽

Cited By ~ 1

Author(s):

Tiwei Zhao ◽

Zenghui Gao ◽

Zhaoyong Huang

Keyword(s):

Graded Modules ◽

Flat Modules ◽

Duality Pairs ◽

Coherent Rings ◽

Flat Cover ◽

Finitely Presented

Let [Formula: see text] be an integer. We introduce the notions of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Then we investigate the properties of these modules by using the properties of special finitely presented graded modules and obtain some equivalent characterizations of [Formula: see text]-gr-coherent rings in terms of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Moreover, we prove that the pairs (gr-[Formula: see text], gr-[Formula: see text]) and (gr-[Formula: see text], gr-[Formula: see text]) are duality pairs over left [Formula: see text]-coherent rings, where gr-[Formula: see text] and gr-[Formula: see text] denote the subcategories of [Formula: see text]-FP-gr-injective left [Formula: see text]-modules and [Formula: see text]-gr-flat right [Formula: see text]-modules respectively. As applications, we obtain that any graded left (respectively, right) [Formula: see text]-module admits an [Formula: see text]-FP-gr-injective (respectively, [Formula: see text]-gr-flat) cover and preenvelope.

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A non-commutative analogue of Costa’s first conjecture

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500073 ◽

2019 ◽

Vol 19 (01) ◽

pp. 2050007

Author(s):

Weiqing Li ◽

Dong Liu

Keyword(s):

Negative Integer ◽

Noetherian Rings ◽

Coherent Ring ◽

Coherent Rings

Let [Formula: see text] and [Formula: see text] be arbitrary fixed integers. We prove that there exists a ring [Formula: see text] such that: (1) [Formula: see text] is a right [Formula: see text]-ring; (2) [Formula: see text] is not a right [Formula: see text]-ring for each non-negative integer [Formula: see text]; (3) [Formula: see text] is not a right [Formula: see text]-ring [Formula: see text]for [Formula: see text], for each non-negative integer [Formula: see text]; (4) [Formula: see text] is a right [Formula: see text]-coherent ring; (5) [Formula: see text] is not a right [Formula: see text]-coherent ring. This shows the richness of right [Formula: see text]-rings and right [Formula: see text]-coherent rings, and, in particular, answers affirmatively a problem posed by Costa in [D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra 22 (1994) 3997–4011.] when the ring in question is non-commutative.

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AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788718000290 ◽

2018 ◽

Vol 107 (02) ◽

pp. 181-198

Author(s):

JAMES GILLESPIE

Keyword(s):

The Other ◽

Homotopy Category ◽

Gorenstein Ring ◽

Model Structure ◽

Projective Modules ◽

Stable Module Category ◽

Stable Module ◽

Coherent Rings ◽

Quillen Equivalent ◽

Compactly Generated

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

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