scholarly journals Generalized lifting modules

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
Yongduo Wang ◽  
Nanqing Ding
Keyword(s):  
Exact Sequence ◽  
Finitely Generated ◽  
Left Module ◽  
Noetherian Module

We introduce the concepts of lifting modules and (quasi-)discrete modules relative to a given left module. We also introduce the notion of SSRS-modules. It is shown that (1) ifMis an amply supplemented module and0→N′→N→N″→0an exact sequence, thenMisN-lifting if and only if it isN′-lifting andN″-lifting; (2) ifMis a Noetherian module, thenMis lifting if and only ifMisR-lifting if and only ifMis an amply supplemented SSRS-module; and (3) letMbe an amply supplemented SSRS-module such thatRad(M)is finitely generated, thenM=K⊕K′, whereKis a radical module andK′is a lifting module.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

scholarly journals Completely Faithful Modules and Self-Injective Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 697-708 ◽  
Author(s):  
Goro Azumaya
Keyword(s):  
Endomorphism Ring ◽  
Finitely Generated ◽  
Left Module

A left module over a ring Λ is called completely faithful if Λ is a sum of those left ideals which are homomorphic images of M. The notion was first introduced by Morita [9], and he proved, among others, the following theorem which plays a basic role in his theory of category-isomorphisms: if a Λ-module M is completely faithful, then M is finitely generated and projective with respect to the endomorphism ring Γ of M and Λ coincides with the endomorphism ring of Λ-module M.

Gelfand-Kirillov Dimension is Exact for Noetherian PI Algebras

1984 ◽  
Vol 27 (2) ◽  
pp. 247-250 ◽  
Author(s):  
T. H. Lenagan
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Finitely Generated ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

AbstractIf O → A → C → B → O is a short exact sequence of finitely generated modules over a Noetherian Pi-algebra then we show that GK(C) = max{GK(A), GK(B)}.

Cohomological Dimension of Generalized Local Cohomology Modules

2008 ◽  
Vol 15 (02) ◽  
pp. 303-308 ◽  
Author(s):  
Jafar Amjadi ◽  
Reza Naghipour
Keyword(s):  
Exact Sequence ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Cohomological Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

The study of the cohomological dimension of algebraic varieties has produced some interesting results and problems in local algebra. Let 𝔞 be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N, the concept of cohomological dimension cd 𝔞(M, N) of M and N with respect to 𝔞 is introduced. If 0 → N' → N → N'' → 0 is an exact sequence of finitely generated R-modules, then it is shown that cd 𝔞(M, N) = max { cd 𝔞(M, N'), cd 𝔞(M, N'')} whenever proj dim M < ∞. Also, if L is a finitely generated R-module with Supp (N/Γ𝔞(N)) ⊆ Supp (L/Γ𝔞(L)), then it is proved that cd 𝔞(M, N) ≤ max { cd 𝔞(M,L), proj dim M}. Finally, as a consequence, a result of Brodmann is improved.

Explicit homotopy equivalences in dimension two

2002 ◽  
Vol 133 (3) ◽  
pp. 411-430 ◽  
Author(s):  
F. E. A. JOHNSON
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Finitely Generated ◽  
Stable Class ◽  
A Finite Group

Let G be a finite group; by an algebraic 2-complex over G we mean an exact sequence of Z[G]-modules of the form:E = (0 → J → E2 → E1 → E0 → Z → 0)where Er is finitely generated free over Z[G] for 0 [les ] r [les ] 2. The module J is determined up to stability by the fact of appearing in such an exact sequence; we denote its stable class by Ω3(Z); E is said to be minimal when rkZ(J) attains the minimum possible value within Ω3(Z).

Neat-phantom and clean-cophantom morphisms

2020 ◽  
pp. 2150172
Author(s):  
Lixin Mao
Keyword(s):  
Exact Sequence ◽  
Projective Resolution ◽  
Injective Envelope ◽  
Finitely Generated ◽  
The Relationship ◽  
Exact Sequences

Let [Formula: see text] be the class of all left [Formula: see text]-modules [Formula: see text] which has a projective resolution by finitely generated projectives. An exact sequence [Formula: see text] of right [Formula: see text]-modules is called neat if the sequence [Formula: see text] is exact for any [Formula: see text]. An exact sequence [Formula: see text] of left [Formula: see text]-modules is called clean if the sequence [Formula: see text] is exact for any [Formula: see text]. We prove that every [Formula: see text]-module has a clean-projective precover and a neat-injective envelope. A morphism [Formula: see text] of right [Formula: see text]-modules is called a neat-phantom morphism if [Formula: see text] for any [Formula: see text]. A morphism [Formula: see text] of left [Formula: see text]-modules is said to be a clean-cophantom morphism if [Formula: see text] for any [Formula: see text]. We establish the relationship between neat-phantom (respectively, clean-cophantom) morphisms and neat (respectively, clean) exact sequences. Also, we prove that every [Formula: see text]-module has a neat-phantom cover with kernel neat-injective and a clean-cophantom preenvelope with cokernel clean-projective.

Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity

1996 ◽  
Vol 119 (3) ◽  
pp. 425-445 ◽  
Author(s):  
D. Kirby ◽  
D. Rees
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Noetherian Ring ◽  
Free Module ◽  
Graded Rings ◽  
Finitely Generated ◽  
Graded Ring ◽  
The Subject ◽  
Integral Equivalence ◽  
Image Position

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

Structure of Some Noetherian Injective Modules

1980 ◽  
Vol 32 (6) ◽  
pp. 1277-1287 ◽  
Author(s):  
B. Sarath
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Commutative Case ◽  
Noetherian Module ◽  
Direct Sum ◽  
Injective Modules ◽  
Semiprime Rings ◽  
Noetherian Modules

The main object of this paper is to study when infective noetherian modules are artinian. This question was first raised by J. Fisher and an example of an injective noetherian module which is not artinian is given in [9]. However, it is shown in [4] that over commutative rings, and over hereditary noetherian P.I rings, injective noetherian does imply artinian. By combining results of [6] and [4] it can be shown that the above implication is true over any noetherian P.I ring. It is shown in this paper that injective noetherian modules are artinian over rings finitely generated as modules over their centers, and over semiprime rings of Krull dimension 1. It is also shown that every injective noetherian module over a P.I ring contains a simple submodule. Since any noetherian injective module is a finite direct sum of indecomposable injectives it suffices to study when such injectives are artinian. IfQis an injective indecomposable noetherian module, thenQcontains a non-zero submoduleQ0such that the endomorphism rings ofQ0and all its submodules are skewfields. Over a commutative ring, such aQ0is simple. In the non-commutative case very little can be concluded, and many of the difficulties seem to arise here.

Ziegler's Indecomposability Criterion

2013 ◽  
Vol 56 (3) ◽  
pp. 564-569
Author(s):  
Ivo Herzog
Keyword(s):  
Noetherian Ring ◽  
Indecomposable Module ◽  
Sufficient Condition ◽  
Finitely Generated ◽  
Left Module ◽  
Indecomposable Modules

Abstract.Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e., ∑-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module RU to have an indecomposable local dual

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