Module-theoretic characterizations of the ring of finite fractions of a commutative ring

Wang, Fang Gui; Zhou, De Chuan; Chen, Dan

doi:10.1216/jca.2022.14.141

Cited by 6 publications

(14 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, R itself is not always a Lucas module. It was proved in [20,Proposition 3.8] that a ring R is a Lucas module if and only if the q-and w-operations on R coincide, if and only if every finitely generated semi-regular ideal is a GV-ideal, if and only if Q 0 (R) = R. For convenience, we say a ring R is a WQ-ring if every finitely generated semi-regular ideal is a GV-ideal. Obviously, a ring R is a DQ-ring if and only if it is both a DW-ring and a WQ-ring.…”

Section: Rings With Sr-w-weak Global Dimensions At Most Onementioning

confidence: 99%

“…Throughout this paper, we always assume R is a commutative ring with identity and T (R) is the total quotient ring of R. Following from [17], an ideal I of R is said to be dense if (0 : R I) := {r ∈ R | Ir = 0} = 0 and be semi-regular if it contains a finitely generated dense sub-ideal. Denote by Q the set of all finitely generated semi-regular ideals of R. Following from [20] that a ring R is called a DQ-ring if Q = {R}. If R is an integral domain, the quotient field K is a very important R-module to study integral domains.…”

Section: Introductionmentioning

confidence: 99%

“…[11, Example 12] Let D = L[X 2 , X 3 , Y ], P = Spec(D)−{ X 2 , X 3 , Y }, B = p∈P K(R/p) and R = D(+)B where L is a field and K(R/p) is the quotient field of R/p. Since Q 0 (R) = R, R is a WQ-ring by[20, Proposition 3.8]. Since every finitely generated R-ideal of the form J(+)B with√ J = X 2 , X 3 , Y is a GV-ideal,R is not a DW-ring.…”

mentioning

confidence: 99%

See 2 more Smart Citations

A homological characterization of Q0-Prüfer v-multiplication rings

Zhang

2022

International Electronic Journal of Algebra

View full text Add to dashboard Cite

Let $R$ be a commutative ring. An $R$-module $M$ is called a semi-regular $w$-flat module if $\Tor_1^R(R/I,M)$ is $\GV$-torsion for any finitely generated semi-regular ideal $I$. In this article, we show that the class of semi-regular $w$-flat modules is a covering class. Moreover, we introduce the semi-regular $w$-flat dimensions of $R$-modules and the $sr$-$w$-weak global dimensions of the commutative ring $R$. Utilizing these notions, we give some homological characterizations of $\WQ$-rings and $Q_0$-\PvMR s.

show abstract

Section: Rings With Sr-w-weak Global Dimensions At Most Onementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A homological characterization of Q0-Prüfer v-multiplication rings

Zhang

2022

International Electronic Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…The notion of Lucas modules was first introduced by Wang et al [9] to give an example of total rings with small finitistic dimension larger than zero, which provided a negative solution to the open question in [1,Problem 1b]. Actually, they showed that if a ring R has small finitistic dimension equal to zero, then R satisfies that every R-module is a Lucas modules (see [9, Theorem 3.9, Proposition 3.10]).…”

Section: Introductionmentioning

confidence: 99%

“…So we will firstly recall the definitions on Lucas modules and q-operations. For more details on this topic, see [9,10,12,15].…”

Section: Introductionmentioning

confidence: 99%

Some remarks on Lucas modules

Zhang¹,

Dai²,

Qi³

2022

Preprint

View full text Add to dashboard Cite

show abstract

Some Characterizations of w‐Noetherian Rings and SM Rings

Zhou

Kim

Zhang

et al. 2022

Journal of Mathematics

Self Cite

View full text Add to dashboard Cite

In this paper, we characterize w -Noetherian rings and SM rings. More precisely, in terms of the u -operation on a commutative ring R , we prove that R is w -Noetherian if and only if the direct limit of r GV -torsion-free injective R -modules is injective and that R is SM, which can be regarded as a regular w -Noetherian ring, if and only if the direct limit of GV -torsion-free (or r GV -torsion-free) reg-injective R -modules is reg-injective. As a by-product of the proof of the second statement, we also obtain that the direct and inverse limits of u -modules are both u -modules and that SM rings are regular w -coherent.

show abstract

Module-theoretic characterizations of the ring of finite fractions of a commutative ring

Cited by 6 publications

References 16 publications

A homological characterization of Q0-Prüfer v-multiplication rings

A homological characterization of Q0-Prüfer v-multiplication rings

Some remarks on Lucas modules

Some Characterizations of w‐Noetherian Rings and SM Rings

Contact Info

Product

Resources

About