scholarly journals n-Coherence Relative to a Hereditary Torsion Theory

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Zhu Zhanmin
Keyword(s):  
Positive Integer ◽  
Torsion Theory ◽  
Hereditary Torsion Theory ◽  
Coherent Rings

Let R be a ring, τ=T,ℱ a hereditary torsion theory of mod-R, and n a positive integer. Then, R is called right τ-n-coherent if every n-presented right R-module is τ,n+1-presented. We present some characterizations of right τ-n-coherent rings, as corollaries, and some characterizations of right n-coherent rings and right τ-coherent rings are obtained.

RELATIVE COHERENCE OF RINGS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250047
Author(s):  
LIXIN MAO ◽  
NANQING DING
Keyword(s):  
Left Ideal ◽  
Torsion Theory ◽  
Hereditary Torsion Theory ◽  
Injective Modules ◽  
Coherent Ring ◽  
Coherent Rings ◽  
Finitely Presented

Let R be a ring and τ a hereditary torsion theory for the category of all left R-modules. A right R-module M is called τ-flat if Tor 1(M, R/I) = 0 for any τ-finitely presented left ideal I. A left R-module N is said to be τ-f-injective in case Ext 1(R/I, N) = 0 for any τ-finitely presented left ideal I. R is called a left τ-coherent ring in case every τ-finitely presented left ideal is finitely presented. τ-coherent rings are characterized in terms of, among others, τ-flat and τ-f-injective modules. Some known results are extended.

scholarly journals Hereditary Torsion Theory Counter Equivalences

1996 ◽  
Vol 183 (1) ◽  
pp. 217-230 ◽  
Author(s):  
R.R. Colby ◽  
K.R. Fuller
Keyword(s):  
Torsion Theory ◽  
Hereditary Torsion Theory

scholarly journals Lattice-ordered modules of quotients

1980 ◽  
Vol 30 (2) ◽  
pp. 243-251 ◽  
Author(s):  
Stuart A. Steinberg
Keyword(s):  
Jacobson Radical ◽  
Torsion Theory ◽  
Finitely Generated ◽  
Hereditary Torsion Theory ◽  
Ring Of Quotients

AbstractLet Q be the ring of quotients of the f-ring R with respect to a positive hereditary torsion theory and suppose Q is a right f-ring. It is shown that if the finitely-generated right ideals of R are principal, then Q is an f-ring. Also, if QR is injective, Q is an f-ring if and only if its Jacobson radical is convex. Moreover, a class of po-rings is introduced (which includes the classes of commutative po-rings and right convex f-rings) over which Q(M) is an f-module for each f-module M.

scholarly journals Change of ring and torsion-theoretic injectivity

2007 ◽  
Vol 75 (1) ◽  
pp. 127-133
Author(s):  
Iuliu Crivei ◽  
Septimiu Crivei ◽  
Ioan Purdea
Keyword(s):  
Ring Homomorphism ◽  
Torsion Theory ◽  
Hereditary Torsion Theory

Let τ be a hereditary torsion theory in R-Mod. Then any ring homomorphism γ: R → S induces in S-Mod a torsion theory σ given by the condition that a left S-module is σ-torsion if and only if it is τ-torsion as a left R-module. We show that if γ: R → S is a ring epimorphism and A is a τ-injective left R-module, then AnnA Ker(γ) is σ-injective as a left S-module. As a consequence, we relate τ-injectivity and σ-injectivity, and we give some applications.

Two applications of Nagata rings and modules

2019 ◽  
Vol 19 (06) ◽  
pp. 2050115
Author(s):  
Fanggui Wang ◽  
Lei Qiao
Keyword(s):  
New York ◽  
Commutative Ring ◽  
Applied Mathematics ◽  
Torsion Theory ◽  
Ideal Theory ◽  
Star Operation ◽  
Text Version ◽  
Commutative Theory ◽  
Coherent Rings ◽  
Relative Invariants

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

scholarly journals Higher derivations on rings and modules

2005 ◽  
Vol 2005 (15) ◽  
pp. 2373-2387 ◽  
Author(s):  
Paul E. Bland
Keyword(s):  
Torsion Theory ◽  
Hereditary Torsion Theory ◽  
Localization Functor ◽  
Higher Derivation

Letτbe a hereditary torsion theory onModRand suppose thatQτ:ModR→ModRis the localization functor. It is shown that for allR-modulesM, every higher derivation defined onMcan be extended uniquely to a higher derivation defined onQτ(M)if and only ifτis a higher differential torsion theory. It is also shown that ifτis a TTF theory andCτ:M→Mis the colocalization functor, then a higher derivation defined onMcan be lifted uniquely to a higher derivation defined onCτ(M).

Extensions of semi-hereditary rings

1977 ◽  
Vol 23 (3) ◽  
pp. 333-339 ◽  
Author(s):  
M. W. Evans
Keyword(s):  
Regular Ring ◽  
Torsion Theory ◽  
Hereditary Torsion Theory ◽  
Left And Right ◽  
Free Modules ◽  
Torsion Free

AbstractHattori (1960) defined a right R-module A to be torsion-free if for all a ∈ A and x ∈ R, ax = 0 implies that there exist elements {x1, x2, …, xn} ⊆ R with xix = 0 for all 1 ≦ i ≦ n and {a1, a2, …, an} ⊆ A such that a = aixi. Left torsion-free is defined similarly. It is shown that for a ring R, these torsion-free modules are the torsion-free class of a hereditary torsion theory, corresponding to a perfect topology, if and only if the left flat epimorphic hull of R is a regular ring which is both left and right torsion-free. A class of right semi-hereditary rings for which the torsion-free modules of Hattori satisfy the above property are found and this class of rings is discussed.

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