scholarly journals Injective hulls of many-sorted ordered algebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1400-1410
Author(s):  
Xia Zhang ◽  
Wen Ma ◽  
Wolfgang Rump
Keyword(s):  
Injective Hull ◽  
Universal Algebras ◽  
Concrete Form ◽  
Ordered Algebras ◽  
Injective Hulls

Abstract This paper is devoted to the study of injectivity for ordered universal algebras. We first characterize injectives in the category $\begin{array}{} \displaystyle {\mathsf{OAL}_{{\it\Sigma}}^{\leqslant}} \end{array}$ of ordered Σ-algebras with lax morphisms as sup-Σ-algebras. Second, we show that every ordered Σ-algebra has an σ⩽-injective hull, and give its concrete form.

Constructing pure injective hulls

1980 ◽  
Vol 45 (3) ◽  
pp. 544-548 ◽  
Author(s):  
Wilfrid Hodges
Keyword(s):  
Abelian Group ◽  
Homomorphic Image ◽  
Injective Hull ◽  
Pure Extension ◽  
The One ◽  
Injective Hulls

Let A be an abelian group and B a pure injective pure extension of A. Then there is a homomorphic image C of B over A which is a pure injective hull of A; C can be constructed by using Zorn's lemma to find a suitable congruence on B. In a paper [4] which greatly generalises this and related facts about pure injectives, Walter Taylor asks (Problem 1.5) whether one can find a “construction” of C which is more concrete than the one mentioned above; he asks also whether the points of C can be explicitly described. In this note I return the answer No.

On Ring Properties of Injective Hulls

1975 ◽  
Vol 18 (2) ◽  
pp. 233-239 ◽  
Author(s):  
N. C. Lang
Keyword(s):  
Regular Ring ◽  
Associative Ring ◽  
Injective Hull ◽  
Von Neumann ◽  
Singular Ideal ◽  
The Right ◽  
Injective Hulls

Let R be an associative ring and denote by the injective hull of the right module RR. If can be endowed with a ring multiplication which extends the existing module multiplication, we say that is a ring and the statement that R is a ring will always mean in this sense.It is known that is a regular ring (in the sense of von Neumann) if and only if the singular ideal of R is zero.

Injective hulls for ordered algebras

2016 ◽  
Vol 76 (3) ◽  
pp. 339-349 ◽  
Author(s):  
Xia Zhang ◽  
Valdis Laan
Keyword(s):  
Ordered Algebras ◽  
Injective Hulls

scholarly journals Parameter-test-ideals of Cohen–Macaulay rings

2008 ◽  
Vol 144 (4) ◽  
pp. 933-948 ◽  
Author(s):  
Mordechai Katzman
Keyword(s):  
Local Cohomology ◽  
Residue Field ◽  
Injective Hull ◽  
Local Cohomology Modules ◽  
D Modules ◽  
Frobenius Map ◽  
Parameter Test ◽  
Computing Parameter ◽  
Injective Hulls ◽  
Cohomology Modules

AbstractWe describe an algorithm for computing parameter-test-ideals in certain local Cohen–Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp’s notion of ‘special ideals’. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article Generators of D-modules in positive characteristic (J. Alvarez-Montaner, M. Blickle and G. Lyubeznik, Math. Res. Lett. 12 (2005), 459–473).

When projective covers and injective hulls are isomorphic

1973 ◽  
Vol 8 (3) ◽  
pp. 471-476 ◽  
Author(s):  
Ann K. Boyle
Keyword(s):  
Projective Cover ◽  
Injective Hull ◽  
Finitely Generated ◽  
Injective Hulls ◽  
Finitely Generated Modules ◽  
Projective Covers

It is shown that rings in which the projective cover and injective hull of cyclic modules are isomorphic are equivalent to uniserial rings. Further, it is shown that rings for which the top and bottom of finitely generated modules are isomorphic also are equivalent to uniserial rings.

Injective Hulls of Torsion Free Modules

1971 ◽  
Vol 23 (6) ◽  
pp. 1094-1101 ◽  
Author(s):  
J. Zelmanowitz
Keyword(s):  
Nonzero Element ◽  
Injective Hull ◽  
Direct Products ◽  
Finitely Generated ◽  
Basic Theorem ◽  
Direct Sum ◽  
Free Modules ◽  
Injective Hulls ◽  
First Time ◽  
Torsion Free

In § 1, we begin with a basic theorem which describes a convenient embedding of a nonsingular left R-module into a complete direct product of copies of the left injective hull of R (Theorem 2). Several applications follow immediately. Notably, the injective hull of a finitely generated nonsingular left R-module is isomorphic to a direct sum of injective hulls of closed left ideals of R (Corollary 4). In particular, when R is left self-injective, every finitely generated nonsingular left R-module is isomorphic to a finite direct sum of injective left ideals (Corollary 6).In § 2, where it is assumed for the first time that rings have identity elements, we investigate more generally the class of left R-modules which are embeddable in direct products of copies of the left injective hull Q of R. Such modules are called torsion free, and can also be characterized by the property that no nonzero element is annihilated by a dense left ideal of R (Proposition 12).

Mutual Infective Hulls

1996 ◽  
Vol 39 (1) ◽  
pp. 68-73 ◽  
Author(s):  
S. K. Jain ◽  
S. R. López-Permouth ◽  
S. Raza Syed
Keyword(s):  
Injective Hull ◽  
Direct Sums ◽  
Injective Hulls

AbstractThe mutual injective hull of an arbitrary family of modules is constructed. Applications to the calculations of quasi-injective and π-injective hulls of direct sums are given.

An Epi-Reflector for Universal Theories

1973 ◽  
Vol 16 (2) ◽  
pp. 167-171 ◽  
Author(s):  
Paul D. Bacsich
Keyword(s):  
Algebraic Closure ◽  
Amalgamation Property ◽  
Injective Hull ◽  
Universal Theory ◽  
Quotient Field ◽  
Injective Hulls

A construction of an epi-reflector by injective hull techniques is given which applies to the class of models of any universal theory with the Amalgamation Property and there yields a weak but functorial type of algebraic closure. Various completions such as the boolean envelope and quotient field constructions are identified as such injective hulls over epimorphic injections. Forms of the Amalgamation Property are also shown to eliminate various pathologies of epimorphisms and equalizers.

MODULES WHICH ARE INVARIANT UNDER AUTOMORPHISMS OF THEIR INJECTIVE HULLS

2012 ◽  
Vol 12 (02) ◽  
pp. 1250159 ◽  
Author(s):  
TSIU-KWEN LEE ◽  
YIQIANG ZHOU
Keyword(s):  
Injective Hull ◽  
Basic Properties ◽  
Injective Modules ◽  
Injective Hulls

A module is defined to be an automorphism-invariant module if it is invariant under automorphisms of its injective hull. Quasi-injective modules and, more generally, pseudo-injective modules are all automorphism-invariant. Here we develop basic properties of these modules, and discuss when an automorphism-invariant module is quasi-injective or injective. Some known results on quasi-injective and pseudo-injective modules are extended.

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