Noncommutative G-hereditary rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500523 ◽

2020 ◽

pp. 2150052

Author(s):

Yunxia Li ◽

Jian Wang ◽

Yuxian Geng ◽

Jiangsheng Hu

Keyword(s):

Commutative Ring ◽

Associative Ring ◽

Partial Answer ◽

Hereditary Ring ◽

Gorenstein Flat ◽

Gorenstein Injective

In this paper, we introduce and study left (right) [Formula: see text]-hereditary rings over any associative ring, and these rings are exactly [Formula: see text]-hereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. As applications, we give a partial answer to the question posed by Bazzoni, Cortés-Izurdiaga and Estrada, and characterize when the character module of a Gorenstein injective left [Formula: see text]-module is Gorenstein flat provided that [Formula: see text] is a left [Formula: see text]-hereditary ring. In addition, some new characterizations of left hereditary rings are given.

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Some conditions for the existence of Gorenstein projective covers and preenvelopes

Journal of Algebra and Its Applications ◽

10.1142/s0219498816501565 ◽

2016 ◽

Vol 15 (08) ◽

pp. 1650156

Author(s):

Bin Yu

Keyword(s):

Commutative Ring ◽

Projective Modules ◽

Gorenstein Projective Modules ◽

Gorenstein Flat ◽

Gorenstein Injective ◽

Projective Covers

In this paper, we investigate the rings over which a module is Gorenstein flat if and only if it is Gorenstein projective. Some examples of such rings are given. We show that over such rings the class of Gorenstein projective modules is covering. We also characterize the rings over which the class of Gorenstein projective modules is preenveloping. As a conclusion, we obtain that a commutative ring is artinian if and only if every module has a Gorenstein projective preenvelope. The existence of pure injective Gorenstein injective preenvelopes over certain rings is also shown.

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Noncommutative G-semihereditary rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500147 ◽

2018 ◽

Vol 17 (01) ◽

pp. 1850014 ◽

Cited By ~ 2

Author(s):

Jian Wang ◽

Yunxia Li ◽

Jiangsheng Hu

Keyword(s):

Associative Ring ◽

Sufficient Condition ◽

Projective Modules ◽

The Third ◽

Flat Modules ◽

Gorenstein Projective Modules ◽

Direct Limits ◽

Gorenstein Flat ◽

Finitely Presented ◽

Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

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Modules with annihilation property

Journal of Algebra and Its Applications ◽

10.1142/s0219498821501267 ◽

2020 ◽

pp. 2150126

Author(s):

Rasul Mohammadi ◽

Ahmad Moussavi ◽

Masoome Zahiri

Keyword(s):

Commutative Ring ◽

Associative Ring ◽

Zero Divisor ◽

Commutative Rings ◽

Finitely Generated ◽

Ideal Formula ◽

Ordered Monoid ◽

Zero Divisors ◽

The Right ◽

Totally Ordered Monoid

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

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Gorenstein Injective and Gorenstein Flat Dimensions Under Base Change

Communications in Algebra ◽

10.1081/agb-120017355 ◽

2003 ◽

Vol 31 (2) ◽

pp. 991-1005 ◽

Cited By ~ 2

Author(s):

Leila Khatami ◽

Siamak Yassemi

Keyword(s):

Base Change ◽

Gorenstein Flat ◽

Gorenstein Injective

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C-Commutativity

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700016542 ◽

1980 ◽

Vol 30 (2) ◽

pp. 252-255 ◽

Cited By ~ 1

Author(s):

T. Cheatham ◽

E. Enochs

Keyword(s):

Finite Number ◽

Commutative Ring ◽

Associative Ring ◽

Nonzero Divisor ◽

Special Case

AbstractAn associative ring R with identity is said to be c-commutative for c ∈ R if a, b ∈ R and ab = c implies ba = c. Taft has shown that if R is c-commutative where c is a central nonzero divisor]can be omitted. We show that in R[x] is h(x)-commutative for any h(x) ∈ R [x] then so is R with any finite number of (commuting) indeterminates adjoined. Examples adjoined. Examples are given to show that R [[x]] need not be c-commutative even if R[x] is, Finally, examples are given to answer Taft's question for the special case of a zero-commutative ring.

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Acyclic Complexes and Gorenstein Rings

Algebra Colloquium ◽

10.1142/s1005386720000474 ◽

2020 ◽

Vol 27 (03) ◽

pp. 575-586

Author(s):

Sergio Estrada ◽

Alina Iacob ◽

Holly Zolt

Keyword(s):

Analogous Result ◽

Gorenstein Ring ◽

Gorenstein Rings ◽

Injective Modules ◽

Flat Modules ◽

Coherent Ring ◽

Acyclic Complexes ◽

Gorenstein Flat ◽

Gorenstein Injective

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

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GORENSTEIN PROJECTIVE, INJECTIVE AND FLAT MODULES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788709000093 ◽

2009 ◽

Vol 87 (3) ◽

pp. 395-407 ◽

Cited By ~ 4

Author(s):

ZHONGKUI LIU ◽

XIAOYAN YANG

Keyword(s):

Homological Algebra ◽

Flat Modules ◽

Gorenstein Flat ◽

Gorenstein Injective

AbstractIn basic homological algebra, projective, injective and flat modules play an important and fundamental role. In this paper, we discuss some properties of Gorenstein projective, injective and flat modules and study some connections between Gorenstein injective and Gorenstein flat modules. We also investigate some connections between Gorenstein projective, injective and flat modules under change of rings.

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