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.

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.

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].

Galois Theory of Essential Extensions of Modules

1972 ◽  
Vol 24 (4) ◽  
pp. 573-579 ◽  
Author(s):  
Sylvia Wiegand
Keyword(s):  
Galois Theory ◽  
Algebraic Closure ◽  
Essential Extension ◽  
Algebraic Extension ◽  
Injective Hull ◽  
Algebraic Systems ◽  
Image Position ◽  
Essential Extensions

The purpose of this paper is to exploit an analogy between algebraic extensions of fields and essential extensions of modules, in which the role of the algebraic closure of a field F is played by the injective hull H(M) of a unitary left R-module M. (The notion of * ‘algebraic’ extensions of general algebraic systems has been studied by Shoda; see, for example [5].)In this analogy, the role of a polynomial p(x) is played by a homomorphism of R-modules(1)which will be called an ideal homomorphism into M. The process of solving the equation p(x) = 0 in F, or in an algebraic extension of F, will be replaced by the process of extending an ideal homomorphism (1) to a homomorphism F* from R into M, or into an essential extension of M.

Essential Extensions of T1-Spaces

1981 ◽  
Vol 24 (2) ◽  
pp. 237-240 ◽  
Author(s):  
Rudolf-E. Hoffmann
Keyword(s):  
Essential Extension ◽  
Injective Hull ◽  
Continuous Maps ◽  
Essential Extensions

AbstractIn the category of T1 -spaces and continuous maps, every space X has a unique maximal essential extension. X has an injective hull iff card X≤1.

scholarly journals Localizations of essential extensions

1988 ◽  
Vol 31 (2) ◽  
pp. 243-247 ◽  
Author(s):  
K. R. Goodearl ◽  
D. A. Jordan
Keyword(s):  
Related Question ◽  
Essential Extension ◽  
Injective Module ◽  
Noncommutative Ring ◽  
Injective Hulls ◽  
Essential Extensions

In an earlier paper [4] we considered the question of whether an injective module E over a noncommutative ring R remains injective after localization with respect to a denominator set X in R. A related question is whether, given an essential extension N of an R-module M, the localization N[X–1] must be an essential extension of M[X–1]. In [1] it is shown that if R is left noetherian and X is central in R, then localization at X preserves both injectivity and essential extensions of left R-modules and, hence, preserves injective hulls and minimal injective resolutions.

scholarly journals Extensions by Simple C*-Algebras: Quasidiagonal Extensions

2005 ◽  
Vol 57 (2) ◽  
pp. 351-399 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Equivalence Classes ◽  
Essential Extension ◽  
Unitary Equivalence ◽  
C Algebras ◽  
Continuous Scale ◽  
Image Position ◽  
Essential Extensions ◽  
Universal Coefficient Theorem ◽  
Unital Simple

AbstractLet A be an amenable separable C*-algebra and B be a non-unital but σ-unital simple C*- algebra with continuous scale. We show that two essential extensions τ1 and τ2 of A by B are approximately unitarily equivalent if and only ifIf A is assumed to satisfy the Universal Coefficient Theorem, there is a bijection fromapproximate unitary equivalence classes of the abovementioned extensions to KL(A,M(B)/B). Using KL(A,M(B)/B), we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.

On essential extensions of direct sums of either injective or projective modules

2014 ◽  
Vol 13 (07) ◽  
pp. 1450038 ◽  
Author(s):  
M. Tamer Koşan ◽  
Truong Cong Quynh
Keyword(s):  
Essential Extension ◽  
Projective Modules ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Hulls ◽  
Essential Extensions

For a ring R, there are classical facts that R is right Noetherian if and only if every direct sum of injective right R-modules is injective, and R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of injective right R-modules. In this paper, we prove that R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of either injective right R-modules or projective right R-modules.

AN ELEMENTARY PROOF OF A THEOREM OF LEE

1991 ◽  
Vol 11 (3) ◽  
pp. 356-360 ◽  
Author(s):  
Jia'an Yan
Keyword(s):  
Elementary Proof

scholarly journals A new elementary proof of a theorem of Minkowski

1926 ◽  
Vol 2 (3) ◽  
pp. 97-99
Author(s):  
Matsusaburô Fujiwara
Keyword(s):  
Elementary Proof
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