Centrally essential rings

2019 ◽  
Vol 29 (3) ◽  
pp. 189-194 ◽  
Author(s):  
Viktor T. Markov ◽  
Askar A. Tuganbaev
Keyword(s):  
Essential Extension

Abstract A centrally essential ring is a ring which is an essential extension of its center (we consider the ring as a module over its center). We give several examples of noncommutative centrally essential rings and describe some properties of centrally essential rings.

A New Construction of the Injective Hull

1968 ◽  
Vol 11 (1) ◽  
pp. 19-21 ◽  
Author(s):  
Isidore Fleischer
Keyword(s):  
Essential Extension ◽  
Injective Hull ◽  
Minimal Cardinality ◽  
New Construction ◽  
Definition Of

The definition of injectivity, and the proof that every module has an injective extension which is a subextension of every other injective extension, are due to R. Baer [B]. An independent proof using the notion of essential extension was given by Eckmann-Schopf [ES]. Both proofs require the p reliminary construction of some injective overmodule. In [F] I showed how the latter proof could be freed from this requirement by exhibiting a set F in which every essential extension could be embedded. Subsequently J. M. Maranda pointed out that F has minimal cardinality. It follows that F is equipotent with the injective hull. Below Icon struct the injective hull by equipping Fit self with a module strucure.

scholarly journals Essential extensions of filters in residuated lattices

2021 ◽  
Author(s):  
Masoud Haveshki
Keyword(s):  
Boolean Algebra ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Essential Extension ◽  
Mv Algebra ◽  
Essential Extensions

Abstract We define the essential extension of a filter in the residuated lattice A associated to an ideal of L(A) and investigate its related properties. We prove the residuated lattice A is a Boolean algebra, G(RL)-algebra or MV -algebra if and only if the essential extension of {1} associated to A \ P is a Boolean filter, G-filter or MV -filter (for all P ∈ SpecA), respectively. Also, some properties of lattice of essential extensions are studied.

scholarly journals Modules with Chain Conditions on S-Closed Submodules

2017 ◽  
Vol 30 (3) ◽  
pp. 227
Author(s):  
Rana Noori Majeed Mohammed
Keyword(s):  
Commutative Ring ◽  
Essential Extension ◽  
Chain Conditions ◽  
Closed Submodule ◽  
Closed Submodules

  Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called  s- closed submodule denoted by  D ≤sc W, if D has   no  proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In  this  paper,  we study  modules which satisfies  the ascending chain  conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.

Injectivity in Equational Classes of Algebras

1972 ◽  
Vol 24 (2) ◽  
pp. 209-220 ◽  
Author(s):  
Alan Day
Keyword(s):  
Abelian Groups ◽  
Essential Extension ◽  
Boolean Algebras ◽  
Equational Class ◽  
Injective Hull ◽  
Fundamental Notion ◽  
First Results ◽  
Initial Results

The concept of injectivity in classes of algebras can be traced back to Baer's initial results for Abelian groups and modules in [1]. The first results in non-module types of algebras appeared when Halmos [14] described the injective Boolean algebras using Sikorski's lemma on extensions of Boolean homomorphisms [19]. In recent years, there have been several results (see references) describing the injective algebras in other particular equational classes of algebras.In [10], Eckmann and Schopf introduced the fundamental notion of essential extension and gave the basic relations that this concept had with injectivity in the equational class of all modules over a given ring. They developed the notion of an injective hull (or envelope) which provided every module with a minimal injective extension or equivalently, a maximal essential extension. In [6] and [9], it was noted that these relationships hold in any equational class with enough injectives.

An Essential Extension with Nonisomorphic Ring Structures II

2007 ◽  
Vol 35 (12) ◽  
pp. 3986-4004 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jae Keol Park ◽  
S. Tariq Rizvi
Keyword(s):  
Essential Extension ◽  
Ring Structures

scholarly journals ON MAXIMAL ESSENTIAL EXTENSIONS OF RINGS

2010 ◽  
Vol 83 (2) ◽  
pp. 329-337 ◽  
Author(s):  
R. R. ANDRUSZKIEWICZ
Keyword(s):  
Elementary Proof ◽  
Essential Extension ◽  
Essential Extensions

AbstractThe main purpose of this paper is to give a new, elementary proof of Flanigan’s theorem, which says that a given ring A has a maximal essential extension ME(A) if and only if the two-sided annihilator of A is zero. Moreover, we discuss the problem of description of ME(A) for a given right ideal A of a ring with an identity.

Metrizability Conditions for Completely Distributive Lattices

1983 ◽  
Vol 26 (4) ◽  
pp. 446-453
Author(s):  
G. Gierz ◽  
J. D. Lawson ◽  
A. R. Stralka
Keyword(s):  
Distributive Lattice ◽  
Essential Extension ◽  
Distributive Lattices ◽  
Interval Topology ◽  
Completely Distributive ◽  
Countable Chain ◽  
Completely Distributive Lattice

AbstractA lattice is said to be essentially metrizable if it is an essential extension of a countable lattice. The main result of this paper is that for a completely distributive lattice the following conditions are equivalent: (1) the interval topology on L is metrizable, (2) L is essentially metrizable, (3) L has a doubly ordergenerating sublattice, (4) L is an essential extension of a countable chain.

pg-Extensions and p-Extensions with applications to C(X)

2015 ◽  
Vol 14 (05) ◽  
pp. 1550068
Author(s):  
Papiya Bhattacharjee ◽  
Michelle L. Knox
Keyword(s):  
Commutative Rings ◽  
Essential Extension ◽  
Ring Extension ◽  
Principal Ideals ◽  
Contraction Map ◽  
Essential Extensions ◽  
Regular Rings

Essential extensions and p-extensions have been studied for commutative rings with identity in various papers, such as [P. Bhattacharjee, M. L. Knox and W. Wm. McGovern, p-Extensions, Proceedings for the OSU-Denison Conference, AMS series Contemporary Mathematics Series (to appear); p-Embeddings, Topology Appl. 160(13) (2013) 1566–1576; R. M. Raphael, Algebraic Extensions of Commutative Regular Rings, Canad. J. Math. 22(6) (1970) 1133–1155]. The present paper applies these concepts to certain subrings of C(X). Moreover, the paper introduces a new ring extension, called a pg-extension, and determines its relation to both essential extension and p-extension. It turns out that the pg-extension R ↪ S induces a well-defined contraction map between principal ideals [Formula: see text] and [Formula: see text].

scholarly journals strongly N-extending Strongly N-extending modules

2021 ◽  
Vol 26 (4) ◽  
Author(s):  
Darya Jabar ◽  
Saad Abdulkadhim Al-Saad
Keyword(s):  
Direct Summand ◽  
Essential Extension ◽  
Other Hand

Relative extending modules and relative (quasi-)continuous modules were introduced and studied by Oshiro as a generalizations of extending modules and (quasi-) continuous respectively.  On other hand, Oshiro, Rizvi and Permouth introduced N-extending and N-(quasi-) continuous modules depending              where N and M are modules.  is closed under submodules, essential extension and isomorphic image. A module M is N-extending if for each submodule A , there is a direct summand B of M such that A is essential in B. Moreover, a module M is strongly extending if every submodule is essential in a stable (equivalently, fully invariant) direct summand of M. In this paper, we introduce and study classes of modules which are proper stronger than that of N-extending modules and N-(quasi-)continuous modules. Many characterizations and properties of these classes are given.

scholarly journals Modules Whose Cyclic Submodules Have Finite Dimension

1976 ◽  
Vol 19 (1) ◽  
pp. 1-6 ◽  
Author(s):  
David Berry
Keyword(s):  
Associative Ring ◽  
Finite Dimension ◽  
Essential Extension ◽  
Injective Hull ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Cyclic Submodules ◽  
Unitary Right

R denotes an associative ring with identity. Module means unitary right R-module. A module has finite Goldie dimension over R if it does not contain an infinite direct sum of nonzero submodules [6]. We say R has finite (right) dimension if it has finite dimension as a right R-module. We denote the fact that M has finite dimension by dim (M)<∞.A nonzero submodule N of a module M is large in M if N has nontrivial intersection with nonzero submodules of M [7]. In this case M is called an essential extension of N. N⊆′M will denote N is essential (large) in M. If N has no proper essential extension in M, then N is closed in M. An injective essential extension of M, denoted I(M), is called the injective hull of M.

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