Injective modules over Mori domains

2003 ◽  
Vol 40 (1-2) ◽  
pp. 33-40
Author(s):  
L. Fuchs
Keyword(s):  
Injective Module ◽  
The Other ◽  
Injective Hull ◽  
Maximum Condition ◽  
Direct Sum ◽  
Injective Modules ◽  
Special Case ◽  
False Claim

Injective modules are considered over commutative domains. It is shown that every injective module admits a decomposition into two summands, where one of the summands contains an essential submodule whose elements have divisorial annihilator ideals, while the other summand contains no element with divisorial annihilator. In the special case of Mori domains (i.e., the divisorial ideals satisfy the maximum condition), the first summand is a direct sum of a S-injective module and a module that has no such summand. The former is a direct sum of indecomposable injectives, while the latter is the injective hull of such a direct sum. Those Mori domains R are characterized for which the injective hull of Q/R is S-injective (Q denotes the field of quotients of R) as strong Mori domains, correcting a false claim in the literature.

Weakly-injective modules over hereditary noetherian prime rings

1995 ◽  
Vol 58 (3) ◽  
pp. 287-297 ◽  
Author(s):  
S. K. Jain ◽  
S. R. López-Permouth
Keyword(s):  
Simple Module ◽  
Injective Module ◽  
Injective Hull ◽  
Prime Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Hereditary Noetherian Prime Ring ◽  
Free Modules ◽  
Torsion Modules ◽  
Torsion Free

AbstractA module M is said to be wealdy-injective if and only if for every finitely generated submodule N of the injective hull E(M) of M there exists a submodule X of E(M), isomorphic to M such that N ⊂ X. In this paper we investigate weakly-injective modules over bounded hereditary noetherian prime rings. In particular we show that torsion-free modules over bounded hnp rings are always wealdy-injective, while torsion modules with finite Goldie dimension are weakly-injective only if they are injective.As an application, we show that weakly-injective modules over bounded Dedekind prime rings have a decomposition as a direct sum of an injective module B, and a module C satisfying that if a simple module S is embeddable in C then the (external) direct sum of all proper submodules of the injective hull of S is also embeddable in C. Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly-injective module has such a decomposition.

scholarly journals Relative injectivity and CS-modules

1994 ◽  
Vol 17 (4) ◽  
pp. 661-666
Author(s):  
Mahmoud Ahmed Kamal
Keyword(s):  
Direct Decomposition ◽  
Injective Hull ◽  
Extra Condition ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Modules ◽  
Semisimple Module ◽  
Continuous Module

In this paper we show that a direct decomposition of modulesM⊕N, withNhomologically independent to the injective hull ofM, is a CS-module if and only ifNis injective relative toMand both ofMandNare CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for quasi-injective modules. But when we confine ourselves to CS-modules we need no conditions on their socles. Then we investigate direct sums of CS-modules which are pairwise relatively inective. We show that every finite direct sum of such modules is a CS-module. This result is known for quasi-continuous modules. For the case of infinite direct sums, one has to add an extra condition. Finally, we briefly discuss modules in which every two direct summands are relatively inective.

scholarly journals Injective Modules Over Non-Artinian Serial Rings

1988 ◽  
Vol 44 (2) ◽  
pp. 242-251 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Affirmative Answer ◽  
Primitive Idempotent ◽  
Injective Hull ◽  
Finitely Generated ◽  
Serial Ring ◽  
Direct Sum ◽  
Injective Modules

AbstractA module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial. When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved.Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has indecomposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is univerial. (c) Every finitely generated left R-module is serial.The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.

scholarly journals On τ-completely decomposable modules

2004 ◽  
Vol 70 (1) ◽  
pp. 163-175 ◽  
Author(s):  
Septimiu Crivei
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Torsion Theory ◽  
Positive Answer ◽  
Essential Extension ◽  
Injective Module ◽  
Injective Hull ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Torsion Theories

For a hereditary torsion theory τ, a moduleAis called τ-completedly decomposable if it is a direct sum of modules that are the τ-injective hull of each of their non-zero submodules. We give a positive answer in several cases to the following generalised Matlis' problem: Is every direct summand of a τ-completely decomposable module still τ-completely decomposable? Secondly, for a commutative Noetherian ringRthat is not a domain, we determine those torsion theories with the property that every τ-injective module is an essential extension of a (τ-injective) τ-completely decomposable module.

scholarly journals On injectiveL-modules

2005 ◽  
Vol 2005 (5) ◽  
pp. 747-754 ◽  
Author(s):  
Paul Isaac
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Injective Module ◽  
Projective Modules ◽  
Module Theory ◽  
Direct Sum ◽  
Injective Modules ◽  
Free Modules

The concepts of free modules, projective modules, injective modules, and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of projective and injectiveL-modules. In this paper, we give an alternate definition for injectiveL-modules and prove that a direct sum ofL-modules is injective if and only if eachL-module in the sum is injective. Also we prove that ifJis an injective module andμis an injectiveL-submodule ofJ, and if0→μ→fv→gη→0is a short exact sequence ofL-modules, thenν≃μ⊕η.

scholarly journals The game of Double-Silver on intervals

2001 ◽  
Vol 26 (8) ◽  
pp. 485-496 ◽  
Author(s):  
Gerald A. Heuer
Keyword(s):  
Measure Zero ◽  
Optimal Strategies ◽  
The Other ◽  
Parameter Plane ◽  
Left Endpoint ◽  
Relative Placement ◽  
Special Case

Silverman's game on intervals was analyzed in a special case by Evans, and later more extensively by Heuer and Leopold-Wildburger, who found that optimal strategies exist (and gave them) quite generally when the intervals have no endpoints in common. They exist in about half the parameter plane when the intervals have a left endpoint or a right endpoint, but not both, in common, and (as Evans had earlier found) exist only on a set of measure zero in this plane if the intervals are identical. The game of Double-Silver, where each player has its own threshold and penalty, is examined. There are several combinations of conditions on relative placement of the intervals, the thresholds and penalties under which optimal strategies exist and are found. The indications are that in the other cases no optimal strategies exist.

Four concepts from “geometrical” stability theory in modules

1992 ◽  
Vol 57 (2) ◽  
pp. 724-740 ◽  
Author(s):  
T. G. Kucera ◽  
M. Prest
Keyword(s):  
Model Theory ◽  
Stability Theory ◽  
The Other ◽  
General Stability ◽  
Injective Modules ◽  
Algebraic Problem

In [H1] Hrushovski introduced a number of ideas concerning the relations between types which have proved to be of importance in stability theory. These relations allow the geometries associated to various types to be connected. In this paper we consider the meaning of these concepts in modules (and more generally in abelian structures). In particular, we provide algebraic characterisations of notions such as hereditary orthogonality, “p -internal” and “p-simple”. These characterisations are in the same spirit as the algebraic characterisations of such concepts as orthogonality and regularity, that have already proved so useful. Of the concepts that we consider, p-simplicity is dealt with in [H3] and the other three concepts in [H2].The descriptions arose out of our desire to develop some intuition for these ideas. We think that our characterisations may well be useful in the same way to others, particularly since our examples are algebraically uncomplicated and so understanding them does not require expertise in the model theory of modules. Furthermore, in view of the increasing importance of these notions, the results themselves are likely to be directly useful in the model-theoretic study of modules and, via abelian structures, in more general stability-theoretic contexts. Finally, some of our characterisations suggest that these ideas may be relevant to the algebraic problem of understanding the structure of indecomposable injective modules.

Silent Victims Revisited: The Special Case of Domestic Violence

PEDIATRICS ◽  
1995 ◽  
Vol 96 (3) ◽  
pp. 511-513
Author(s):  
Barry Zuckerman ◽  
Marilyn Augustyn ◽  
Betsy McAlister Groves ◽  
Steven Parker
Keyword(s):  
Domestic Violence ◽  
Community Violence ◽  
Clinical Experience ◽  
Behavioral Problems ◽  
The Other ◽  
Media Attention ◽  
Witness Violence ◽  
Primary Focus ◽  
Special Case ◽  
At Home

In a commentary published previously, we communicated our concern regarding the plight of children who witness violence.1 Research suggests that children who witness violence suffer significant psychologic and behavioral problems that interfere with their ability to function in school, at home, and with peers. The primary focus of that commentary was children who witnessed community violence. Our ongoing clinical experience, heightened by media attention on domestic violence, including the O.J. Simpson case, leads us to revisit silent victims with a sole focus on those children who witness domestic violence. Domestic violence is a particularly devastating event for a child who, in the presence of danger, typically turns to a parent for protection and for whom there is no comfort or security if one parent is the perpetrator of violence, and the other is a terrified victim.

Psq-injective modules and rings

2021 ◽  
pp. 2250101
Author(s):  
Avanish Kumar Chaturvedi ◽  
Sandeep Kumar
Keyword(s):  
Injective Module ◽  
Injective Modules

For any two right [Formula: see text]-modules [Formula: see text] and [Formula: see text], [Formula: see text] is said to be a ps-[Formula: see text]-injective module if, any monomorphism [Formula: see text] can be extended to [Formula: see text]. Also, [Formula: see text] is called psq-injective if [Formula: see text] is a ps-[Formula: see text]-injective module. We discuss some properties and characterizations in terms of psq-injective modules.

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

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