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 Uniqueness of the maximal ideal of operators on the ℓp-sum of ℓ∞n (n ∈ ) for 1 < p < ∞

2016 ◽  
Vol 160 (3) ◽  
pp. 413-421 ◽  
Author(s):  
TOMASZ KANIA ◽  
NIELS JAKOB LAUSTSEN
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
American Mathematical Society ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Unique Maximal Ideal

AbstractA recent result of Leung (Proceedings of the American Mathematical Society, 2015) states that the Banach algebra ℬ(X) of bounded, linear operators on the Banach space X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓ1 contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓp and X = (⊕n∈$\mathbb{N}$ ℓ1n)ℓp whenever p ∈ (1, ∞).

Ordinary Singularities of Algebraic Curves

1981 ◽  
Vol 24 (4) ◽  
pp. 423-431 ◽  
Author(s):  
Ferruccio Orecchia
Keyword(s):  
Singular Point ◽  
Vector Space ◽  
Local Ring ◽  
Maximal Ideal ◽  
Tangent Cone ◽  
Algebraic Curves ◽  
Vector Spaces ◽  
Generic Position

AbstractLet A be the local ring at a singular point p of an algebraic reduced curve. Let M (resp. Ml,..., Mh) be the maximal ideal of A (resp. of Ā). In this paper we want to classify ordinary singularities p with reduced tangent cone: Spec(G(A)). We prove that G(A) is reduced if and only if: p is an ordinary singularity, and the vector spaces span the vector space . If the points of the projectivized tangent cone Proj(G(A)) are in generic position then p is an ordinary singularity if and only if G(A) is reduced. We give an example which shows that the preceding equivalence is not true in general.

scholarly journals Generalized periodic and generalized Boolean rings

2001 ◽  
Vol 26 (8) ◽  
pp. 457-465
Author(s):  
Howard E. Bell ◽  
Adil Yaqub
Keyword(s):  
Boolean Ring ◽  
Boolean Rings

We prove that a generalized periodic, as well as a generalized Boolean, ring is either commutative or periodic. We also prove that a generalized Boolean ring with central idempotents must be nil or commutative. We further consider conditions which imply the commutativity of a generalized periodic, or a generalized Boolean, ring.

scholarly journals Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

2021 ◽  
pp. 137-154
Author(s):  
Hezron Saka Were ◽  
Maurice Oduor Owino ◽  
Moses Ndiritu Gichuki
Keyword(s):  
Maximal Ideal ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unique Maximal Ideal

In this paper, R is considered a completely primary finite ring and Z(R) is its subset of all zero divisors (including zero), forming a unique maximal ideal. We give a construction of R whose subset of zero divisors Z(R) satisfies the conditions (Z(R))5 = (0); (Z(R))4 ̸= (0) and determine the structures of the unit groups of R for all its characteristics.

scholarly journals The ideal structure of idempotent-generated transformation semigroups

1985 ◽  
Vol 28 (3) ◽  
pp. 319-331 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Boolean Ring ◽  
Ideal Structure ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Ideal

Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.

scholarly journals Near-rings of compatible functions

1980 ◽  
Vol 23 (1) ◽  
pp. 87-95 ◽  
Author(s):  
Günter Pilz
Keyword(s):  
Vector Space ◽  
Structure Theory ◽  
Polynomial Functions ◽  
Affine Transformations ◽  
Affine Completeness ◽  
Congruence Relations

SummaryIn this paper we study near-rings of functions on Ω-groups which are compatible with all congruence relations. Polynomial functions, for instance, are of this type. We employ the structure theory for near-rings to get results for the theory of compatible and polynomial functions (affine completeness, etc.). For notations and results concerning near-rings see e.g. (10). However, we review briefly some terminology from there. (N, +,.) is a near-ring if (N, +) is a group and . is associative and right distributive over +. For instance, M(A): = (AA, +, °) is a near-ring for any group (A, +) (° is composition). If N is a near-ring then N0: = {n ∈ N/n0 = 0}. A group (Γ, +) is an N-group (we write NΓ) if a “product” ny is defined with (n + n‛)γ = nγ + n‛γ and (nn‛)γ = n(n‛γ). Ideals of near-rings and N-groups are kernels of (N-) homomorphisms. If Γ is a vector-space, Maff (Γ) is the near-ring of all affine transformations on Γ. N is 2-primitive on NΓ if NΓ is non-trivial, faithful and without proper N-subgroups. The (2-) radical and (2-) semisimplicity are defined similarly to the ring case.

A Characterization of Implicative Boolean Rings

1953 ◽  
Vol 5 ◽  
pp. 465-469 ◽  
Author(s):  
A. H. Copeland ◽  
Frank Harary
Keyword(s):  
Boolean Algebra ◽  
Cross Product ◽  
Boolean Ring ◽  
Conditional Probabilities ◽  
Boolean Operations ◽  
Boolean Rings ◽  
The Cross

In the theory of probability, the conditional can be treated by an operation analogous to division. Many properties of the conditional can best be studied by means of the corresponding multiplication (called the cross-product). An implicative Boolean ring is defined [2] in terms of a cross-product and the usual Boolean operations. The cross-product is the only device yet known in which the events corresponding to conditional probabilities are themselves elements of the Boolean ring. The fact that such advice was not introduced by Boole is probably the reason why Boolean algebra has been very little used in the theory of probability, although probability was one of the principal applications which Boole had in mind.

Properties of Quotient Rings

1972 ◽  
Vol 24 (6) ◽  
pp. 1122-1128 ◽  
Author(s):  
S. Page
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Ring Of Quotients ◽  
Quotient Rings ◽  
Special Case ◽  
Classical Ring ◽  
Unique Maximal Ideal

In [1; 2 ; 7] Gabriel, Goldman, and Silver have introduced the notion of a localization of a ring which generalizes the usual notion of a localization of a commutative ring at a prime. These rings may not be local in the sense of having a unique maximal ideal. If we are to obtain information about a ring R from one of its localizations, Qτ (R) say, it seems reasonable that Qτ(R) be a tractable ring. This, of course, is what Goldie, Jans, and Vinsonhaler [4; 3; 8] did in the special case for Q(R) the classical ring of quotients.

The Supremum of a Family of Additive Functions

1952 ◽  
Vol 4 ◽  
pp. 463-479 ◽  
Author(s):  
Israel Halperin
Keyword(s):  
Vector Space ◽  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Boolean Ring ◽  
Additive Functions ◽  
Semi Group ◽  
Additive Functionals ◽  
Banach Theorem ◽  
Necessary And Sufficient ◽  
Natural Way

Any system S in which an addition is defined for some, but not necessarily all, pairs of elements can be imbedded in a natural way in a commutative semi-group G, although different elements in S need not always determine different elements in G (see §2). Theorem 2.1 gives necessary and sufficient conditions in order that a functional p(x) on S can be represented as the su prémuni of some family of additive functionals on S, and one such set of conditions is in terms of possible extensions of p(x) to G. This generalizes the case with 5 a Boolean ring treated by Lorentz [4], Lorentz imbeds the Boolean ring in a vector space and this could be done for the general S; but we prefer to imbed S in a commutative semi-group and to give a proof (see § 1) generalizing the classical Hahn-Banach theorem to the case of an arbitrary commutative semigroup.

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