scholarly journals Some results on coherent rings II

1967 ◽  
Vol 8 (2) ◽  
pp. 123-126 ◽  
Author(s):  
Morton E. Harris
Keyword(s):  
Finitely Generated ◽  
Basic Theorem ◽  
Coherent Ring ◽  
Coherent Rings ◽  
Finitely Presented

According to Bourbaki [1, pp. 62–63, Exercise 11], a left (resp. right) A-module M is said to be pseudo-coherent if every finitely generated submodule of M is finitely presented, and is said to be coherent if it is both pseudo-coherent and finitely generated. This Bourbaki reference contains various results on pseudo-coherent and coherent modules. Then, in [1, p. 63, Exercise 12], a ring which as a left (resp. right) module over itself is coherent is said to be a left (resp. right) coherent ring, and various results on and examples of coherent rings are presented. The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3].

Commutative rings and modules that are Nil*-coherent or special Nil*-coherent

2017 ◽  
Vol 16 (10) ◽  
pp. 1750187 ◽  
Author(s):  
Karima Alaoui Ismaili ◽  
David E. Dobbs ◽  
Najib Mahdou
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Basic Properties ◽  
Unital Ring ◽  
Reduced Ring ◽  
Coherent Ring ◽  
Ring Extensions ◽  
Coherent Rings ◽  
Finitely Presented

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.

J-COHERENT RINGS

2009 ◽  
Vol 08 (02) ◽  
pp. 139-155 ◽  
Author(s):  
NANQING DING ◽  
YUANLIN LI ◽  
LIXIN MAO
Keyword(s):  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Flat Modules ◽  
Coherent Ring ◽  
Necessary And Sufficient ◽  
Coherent Rings ◽  
Finitely Presented

Let R be a ring. Recall that a left R-module M is coherent if every finitely generated submodule of M is finitely presented. R is a left coherent ring if the left R-module RR is coherent. In this paper, we say that R is left J-coherent if its Jacobson radical J(R) is a coherent left R-module. J-injective and J-flat modules are introduced to investigate J-coherent rings. Necessary and sufficient conditions for R to be left J-coherent are given. It is shown that there are many similarities between coherent and J-coherent rings. J-injective and J-flat dimensions are also studied.

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 Torsion theories and coherent rings

1973 ◽  
Vol 8 (2) ◽  
pp. 233-239 ◽  
Author(s):  
J.M. Campbell
Keyword(s):  
Direct Product ◽  
Torsion Theory ◽  
Finitely Generated ◽  
Coherent Ring ◽  
Ring Of Quotients ◽  
Torsion Theories ◽  
Coherent Rings ◽  
Finitely Related

Chase has given several characterizations of a right coherent ring, among which are: every direct product of copies of the ring is left-flat; and every finitely generated submodule of a free right module is finitely related. We extend his results to obtain conditions for the ring of quotients of a ring with respect to a torsion theory to be coherent.

C-coherent rings, C-semihereditary rings and C-regular rings

2013 ◽  
Vol 50 (4) ◽  
pp. 491-508 ◽  
Author(s):  
Zhanmin Zhu
Keyword(s):  
Direct Summand ◽  
Finitely Generated ◽  
Coherent Rings ◽  
Finitely Presented ◽  
Regular Rings ◽  
Semihereditary Rings

Let C be a class of some finitely presented left R-modules. A left R-module M is called C-injective, if ExtR1(C, M) = 0 for each C ∈ C. A right R-module M is called C-flat, if Tor1R(M, C) = 0 for each C ∈ C. A ring R is called C-coherent, if every C ∈ C is 2-presented. A ring R is called C-semihereditary, if whenever 0 → K → P → C → 0 is exact, where C ∈ C and P is finitely generated projective and K is finitely generated, then K is also projective. A ring R is called C-regular, if whenever P/K ∈ C, where P is finitely generated projective and K is finitely generated, then K is a direct summand of P. Using the concepts of C-injectivity and C-flatness of modules, we present some characterizations of C-coherent rings, C-semihereditary rings, and C-regular rings.

scholarly journals RATIONAL POLYHEDRA AND PROJECTIVE LATTICE-ORDERED ABELIAN GROUPS WITH ORDER UNIT

2012 ◽  
Vol 14 (03) ◽  
pp. 1250017 ◽  
Author(s):  
LEONARDO CABRER ◽  
DANIELE MUNDICI
Keyword(s):  
Abelian Group ◽  
Algebraic Topology ◽  
Lattice Structure ◽  
Order Unit ◽  
Finitely Generated ◽  
Lattice Order ◽  
Translation Invariant ◽  
Projective Lattice ◽  
Rational Polyhedra ◽  
Finitely Presented

An ℓ-groupG is an abelian group equipped with a translation invariant lattice-order. Baker and Beynon proved that G is finitely generated projective if and only if it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e. an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. Unital ℓ-homomorphisms between unital ℓ-groups are group homomorphisms that also preserve the order unit and the lattice structure. A unital ℓ-group (G, u) is projective if whenever ψ : (A, a) → (B, b) is a surjective unital ℓ-homomorphism and ϕ : (G, u) → (B, b) is a unital ℓ-homomorphism, there is a unital ℓ-homomorphism θ : (G, u) → (A, a) such that ϕ = ψ ◦ θ. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (à la Whitehead) is combined in this paper with the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.

Relative FP-gr-injective and gr-flat modules

2018 ◽  
Vol 28 (06) ◽  
pp. 959-977 ◽  
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.

A non-commutative analogue of Costa’s first conjecture

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.

scholarly journals Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

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