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MODULES SATISFYING CERTAIN CHAIN CONDITIONS AND THEIR ENDOMORPHISMS
Wang, Fanggui,Kim, Hwankoo Korean Mathematical Society 2015 대한수학회보 Vol.52 No.2
In this paper, we characterize w-Noetherian modules in terms of polynomial modules and w-Nagata modules. Then it is shown that for a finite type w-module M, every w-epimorphism of M onto itself is an isomorphism. We also define and study the concepts of w-Artinian modules and w-simple modules. By using these concepts, it is shown that for a w-Artinian module M, every w-monomorphism of M onto itself is an isomorphism and that for a w-simple module M, $End_RM$ is a division ring.
MODULES SATISFYING CERTAIN CHAIN CONDITIONS AND THEIR ENDOMORPHISMS
Fanggui Wang,김환구 대한수학회 2015 대한수학회보 Vol.52 No.2
In this paper, we characterize w-Noetherian modules in terms of polynomial modules and w-Nagata modules. Then it is shown that for a finite type w-module M, every w-epimorphism of M onto itself is an isomorphism. We also define and study the concepts of w-Artinian modules and w-simple modules. By using these concepts, it is shown that for a w-Artinian module M, every w-monomorphism of M onto itself is an isomorphism and that for a w-simple module M, EndRM is a division ring.
A homological characterization of Krull domains
Fanggui Wang,De Chuan Zhou 대한수학회 2018 대한수학회보 Vol.55 No.2
Let $R$ be a commutative ring. In this paper, the $w$-projective Basis Lemma for $w$-projective modules is given. Then it is shown that for a domain, nonzero $w$-projective ideals and nonzero $w$-invertible ideals coincide. As an application, it is proved that $R$ is a Krull domain if and only if every submodule of finitely generated projective modules is $w$-projective.
w-INJECTIVE MODULES AND w-SEMI-HEREDITARY RINGS
Fanggui Wang,김환구 대한수학회 2014 대한수학회지 Vol.51 No.3
Let R be a commutative ring with identity. An R-module M is said to be w-projective if Ext1R(M,N) is GV-torsion for any torsion-free w-module N. In this paper, we define a ring R to be w-semi-hereditary if every finite type ideal of R is w-projective. To characterize w-semi- hereditary rings, we introduce the concept of w-injective modules and study some basic properties of w-injective modules. Using these concepts, we show that R is w-semi-hereditary if and only if the total quotient ring T(R) of R is a von Neumann regular ring and R m is a valuation domain for any maximal w-ideal m of R. It is also shown that a connected ring R is w-semi-hereditary if and only if R is a Prüfer v-multiplication domain.
THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS
WANG, FANGGUI,QIAO, LEI Korean Mathematical Society 2015 대한수학회보 Vol.52 No.4
In this paper, we introduce and study the w-weak global dimension w-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a $Pr\ddot{u}fer$ v-multiplication domain if and only if w-w.gl.dim(R) ${\leq}1$. We also show that there is a large class of domains in which Hilbert's syzygy Theorem for the w-weak global dimension does not hold. Namely, we prove that if R is an integral domain (but not a field) for which the polynomial ring R[x] is w-coherent, then w-w.gl.dim(R[x]) = w-w.gl.dim(R).
w-INJECTIVE MODULES AND w-SEMI-HEREDITARY RINGS
Wang, Fanggui,Kim, Hwankoo Korean Mathematical Society 2014 대한수학회지 Vol.51 No.3
Let R be a commutative ring with identity. An R-module M is said to be w-projective if $Ext\frac{1}{R}$(M,N) is GV-torsion for any torsion-free w-module N. In this paper, we define a ring R to be w-semi-hereditary if every finite type ideal of R is w-projective. To characterize w-semi-hereditary rings, we introduce the concept of w-injective modules and study some basic properties of w-injective modules. Using these concepts, we show that R is w-semi-hereditary if and only if the total quotient ring T(R) of R is a von Neumann regular ring and $R_m$ is a valuation domain for any maximal w-ideal m of R. It is also shown that a connected ring R is w-semi-hereditary if and only if R is a Pr$\ddot{u}$fer v-multiplication domain.
THE ω-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS
Fanggui Wang,Lei Qiao 대한수학회 2015 대한수학회보 Vol.52 No.4
In this paper, we introduce and study the ω-weak global dimension ω-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a Pr¨ufer v-multiplication domain if and only if ω-w.gl.dim(R) ≤ 1. We also show that there is a large class of domains in which Hilbert’s syzygy Theorem for the ω-weak global dimension does not hold. Namely, we prove that if R is an integral domain (but not a field) for which the polynomial ring R[x] is ω-coherent, then ω-w.gl.dim(R[x]) = ω-w.gl.dim(R).
Localization of Injective modules over ω-Noetherian rings
김환구,Fanggui Wang 대한수학회 2013 대한수학회보 Vol.50 No.2
We give some characterizations of injective modules over ω-Noetherian rings. It is also shown that each localization of a GV-torsion-free injective module over a ω-Noetherian ring is injective.
LOCALIZATION OF INJECTIVE MODULES OVER ω-NOETHERIAN RINGS
Kim, Hwankoo,Wang, Fanggui Korean Mathematical Society 2013 대한수학회보 Vol.50 No.2
We give some characterizations of injective modules over ${\omega}$-Noetherian rings. It is also shown that each localization of a GV-torsion-free injective module over a ${\omega}$-Noetherian ring is injective.
ω-MODULES OVER COMMUTATIVE RINGS
Huayu Yin,Fanggui Wang,Xiaosheng Zhu,Youhua Chen 대한수학회 2011 대한수학회지 Vol.48 No.1
Let R be a commutative ring and let M be a GV -torsionfree R-module. Then M is said to be a w-module if Ext<수식> for any J ∈ GV (R), and the ω-envelope of M is defined by Mω = {x ∈E(M)ㅣJx ⊆ M for some J ∈ GV (R)}. In this paper, ω-modules over commutative rings are considered, and the theory of ω-operations is de-veloped for arbitrary commutative rings. As applications, we give some characterizations of ω-Noetherian rings and Krull rings.