The lattice of ideals of an object of a category. I

1966 ◽  
pp. 163-179
Author(s):  
E. G. Šul′geĭfer
Keyword(s):  
Lattice Of Ideals

scholarly journals Corrigendum to "the Lattice of Ideals Of The Nearring of Coset Preserving Functions"

2003 ◽  
Vol 26 (2) ◽  
pp. 141-145 ◽  
Author(s):  
Alan G. Cannon ◽  
Lucyna Kabza
Keyword(s):  
Lattice Of Ideals

Boolean Near-Rings

1969 ◽  
Vol 12 (3) ◽  
pp. 265-273 ◽  
Author(s):  
James R. Clay ◽  
Donald A. Lawver
Keyword(s):  
Vector Space ◽  
Maximal Ideal ◽  
Boolean Ring ◽  
Affine Transformations ◽  
Isomorphism Classes ◽  
Boolean Rings ◽  
Lattice Of Ideals ◽  
Unique Maximal Ideal

In this paper we introduce the concept of Boolean near-rings. Using any Boolean ring with identity, we construct a class of Boolean near-rings, called special, and determine left ideals, ideals, factor near-rings which are Boolean rings, isomorphism classes, and ideals which are near-ring direct summands for these special Boolean near-rings.Blackett [6] discusses the near-ring of affine transformations on a vector space where the near-ring has a unique maximal ideal. Gonshor [10] defines abstract affine near-rings and completely determines the lattice of ideals for these near-rings. The near-ring of differentiable transformations is seen to be simple in [7], For near-rings with geometric interpretations, see [1] or [2].

scholarly journals On the relation of a distributive lattice to its lattice of ideals

1972 ◽  
Vol 7 (3) ◽  
pp. 377-385 ◽  
Author(s):  
Herbert S. Gaskill
Keyword(s):  
Distributive Lattice ◽  
Related Result ◽  
Distributive Lattices ◽  
Relationship Of ◽  
The Relationship ◽  
Finite Distributive Lattices ◽  
Lattice Of Ideals

In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

Embedding finite lattices into the ideals of computably enumerable turing degrees

2001 ◽  
Vol 66 (4) ◽  
pp. 1791-1802
Author(s):  
William C. Calhoun ◽  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Lattices ◽  
Necessary And Sufficient ◽  
Lattice Of Ideals ◽  
Computably Enumerable

Abstract.We show that the lattice L20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (ℐ), We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π2 necessary and sufficient condition for embedding a finite pseudolattice into ℐ.

Symmetric ring property on nil ideals

2018 ◽  
Vol 17 (01) ◽  
pp. 1850013
Author(s):  
Tai Keun Kwak ◽  
Yang Lee ◽  
Zhelin Piao ◽  
Young Joo Seo
Keyword(s):  
Ideal Theory ◽  
Ring Theory ◽  
Noncommutative Rings ◽  
Commutative Ideal ◽  
Ring Extensions ◽  
Lattice Of Ideals

The usual commutative ideal theory was extended to ideals in noncommutative rings by Lambek, introducing the concept of symmetric. Camillo et al. naturally extended the study of symmetric ring property to the lattice of ideals, defining the new concept of an ideal-symmetric ring. This paper focuses on the symmetric ring property on nil ideals, as a generalization of an ideal-symmetric ring. A ring [Formula: see text] will be said to be right (respectively, left) nil-ideal-symmetric if [Formula: see text] implies [Formula: see text] (respectively, [Formula: see text]) for nil ideals [Formula: see text] of [Formula: see text]. This concept generalizes both ideal-symmetric rings and weak nil-symmetric rings in which the symmetric ring property has been observed in some restricted situations. The structure of nil-ideal-symmetric rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

Twisted Brauer monoids

2018 ◽  
Vol 148 (4) ◽  
pp. 731-750 ◽  
Author(s):  
Igor Dolinka ◽  
James East
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Singular Part ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Generating Set ◽  
Idempotent Rank ◽  
Singular Ideal ◽  
Necessary And Sufficient ◽  
Lattice Of Ideals

We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

scholarly journals The lattice of ideals of MR(R2)Ra commutative pir

1992 ◽  
Vol 52 (3) ◽  
pp. 368-382 ◽  
Author(s):  
C. J. Maxson ◽  
L. Van Wyk
Keyword(s):  
Ideal Ring ◽  
Lattice Of Ideals

AbstractIn this paper we characterize the ideals of the centralizer near-ring N = MR(R2), where R is a commutative principle ideal ring. The characterization is used to determine the radicals Jυ(N) and the quotient structures N/ Jv(N), v = 0, 1, 2.

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