Boolean Rings Research Articles

The lattice of ideals of certain rings

Let A be a unitary ring and let (I(A),⊆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathbf {I(A)},\\subseteq )$$\\end{document} be the lattice of ideals of the ring A. In this article, we will study the property of the lattice (I(A),⊆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathbf {I(A)},\\subseteq )$$\\end{document} to be Noetherian or not, for various types of rings A. In the last section of the article, we study certain rings that are not Boolean rings, not fields, but all their ideals are idempotent.

Remarks on rings with u − 1 quasi-nilpotent for each unit u

This paper is concerned with the investigation of those rings [Formula: see text] whose invertible elements are sums [Formula: see text] for an element [Formula: see text] such that [Formula: see text] is invertible for all invertible [Formula: see text] with [Formula: see text]. UJ-rings and UQ-rings (and hence UU-rings) are examples of such rings. We prove that such rings are Dedekind finite, and for semisimple rings [Formula: see text], [Formula: see text] In case [Formula: see text] is also a semipotent ring, it is shown that [Formula: see text] is a Boolean ring.

Complemented subsets and Boolean-valued, partial functions

We study the two algebras of complemented subsets that were introduced in the constructive development of the Daniell approach to measure and integration within Bishop-style constructive mathematics. We present their main properties both for the so-called here categorical complemented subsets and for the extensional complemented subsets. We translate constructively the classical bijection between subsets and Boolean-valued, total functions by establishing a bijection between complemented subsets (categorical or extensional) and Boolean-valued, partial functions (categorical or extensional). The role of Myhill’s axiom of non-choice in the equivalence between categorical and extensional subsets is discussed. We introduce swap algebras of type (I) and (II) as an abstract version of Bishop’s algebras of complemented subsets of type (I) and (II), respectively, and swap rings as an abstract version of the structure of Boolean-valued partial functions on a set. Our examples of swap algebras and swap rings together with the included here results indicate that their theory is a certain generalisation of the theory of Boolean algebras and Boolean rings, a fact which we find interesting both from a constructive and a classical point of view.

Reliability Function and Boolean Ring

Reliability functions are fundamental mathematical models in engineering reliability theory, giving the probability that a system or component functions over a given time period. Boolean rings are algebraic structures with addition, multiplication, and complementation operations that follow certain axioms. This paper shows that collections of reliability functions form Boolean rings under natural definitions of the Boolean operators. Although reliability theory and Boolean rings are mature subjects, the formal connection between them has not been fully recognized. Establishing this link allows the extensive body of results from Boolean ring theory to be applied in analyzing reliability functions. The Boolean perspective leads to simplified calculations and new reliability theorems. We provide mathematical background on reliability functions and Boolean rings before presenting the formal proof that reliability functions satisfy Boolean ring properties. Several examples demonstrate how the Boolean viewpoint yields insight into reliability calculations, system structure functions, and other areas. The relationships revealed here unify reliability theory and Boolean algebra, while offering opportunities for further theoretical developments in both fields. This work elucidates the underlying Boolean structure inherent in reliability modeling.

Spectra of zero-divisor graphs of finite reduced rings

Let [Formula: see text] be a finite reduced ring with [Formula: see text] maximal ideals [Formula: see text] and let [Formula: see text] be the zero-divisor graph associated to [Formula: see text] The class of rings [Formula: see text] contains the Boolean rings as a subclass. When [Formula: see text] for all [Formula: see text] where [Formula: see text] is a finite field, we associate two [Formula: see text] sized matrices [Formula: see text] and [Formula: see text] to the graph [Formula: see text] having combinatorial entries and use these matrices to determine the spectrum of this graph. More precisely, we show that every eigenvalue of [Formula: see text] and of [Formula: see text] is an eigenvalue of [Formula: see text] To do this, we give a recursive description of the adjacency matrix of this graph and also exhibit its equitable partition. This is used in computing the determinant, rank and nullity of the adjacency matrix. Further, we propose that the eigenvalues of [Formula: see text] [Formula: see text] and the eigenvalue [Formula: see text] exhaust all the eigenvalues of [Formula: see text]

Eigenvalues of zero-divisor graphs, Catalan numbers, and a decomposition of eigenspaces

ABSTRACT A combinatorial approach is given to compute bases for eigenspaces of zero-divisor graphs of finite Boolean rings. A commutative monoid S of graphs is shown to contain a cyclic submonoid U that determines values of the entries of basis elements, while the members of its complement S ∖ U encode the supports of these elements. Furthermore, every member of S ∖ U is associated with a Catalan-triangle number, which counts the number of basis elements whose supports are determined by the given member. This is established by using a combinatorial interpretation of Catalan-triangle numbers to produce linearly independent sets of eigenvectors.

On the induced partial action of a quotient group and a structure theorem for a partial Galois extension

Let [Formula: see text] be a ring with a partial action [Formula: see text] of a finite group [Formula: see text]. We determine when a quotient group of [Formula: see text] gives rise to a partial action induced by [Formula: see text] on a subring of [Formula: see text]. As an application, we show that if [Formula: see text] is an [Formula: see text]-partial Galois extension and [Formula: see text] is a normal subgroup of [Formula: see text], then under certain conditions [Formula: see text] under the partial action of [Formula: see text] induced by [Formula: see text], denoted by [Formula: see text], is a partial Galois extension. This result was just shown recently by Bagio et al., applying globalization to derive a partial action of [Formula: see text] on [Formula: see text], totally different from the one presented in this paper arising from a Boolean ring generated by certain idempotents. We further show in either type of induced partial action of a quotient group that under certain conditions [Formula: see text] is a DeMeyer–Kanzaki [Formula: see text]-partial Galois extension whenever [Formula: see text] is an [Formula: see text]-partial Galois Azumaya extension. A structure theorem for a partial Galois extension is also presented, namely, every partial Galois extension can be decomposed as a direct sum of Galois extensions and possibly a trivial partial Galois extension of type II.

Fuzzy Soft Boolean Rings

This article introduces the idea of (fuzzy soft Boolean rings) FSBRs and investigates their algebraic properties. The concepts of fuzzy soft ideals (FSIs) of FSBRs, and idealistic fuzzy soft Boolean rings (IFSBRs) are then defined and discussed.

Induced subgraphs of zero-divisor graphs

The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with a and b adjacent if ab=0. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph, and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the 2-dimensional dot product graph.

On ring-like event systems in quantum logic

A class of ring-like structures of events (RLSEs) is studied that generalizes Boolean rings. Quantum logics represented by orthomodular lattices are characterized within this class and the correspondence between Boolean algebras and Boolean rings is enlarged to orthomodular lattices. The structure of RLSEs and various subclasses is analyzed and classical logics are especially identified. Moreover, sets of numerical events within different contexts of physical problems are described. A numerical event is defined as a function [Formula: see text] from a set [Formula: see text] of states of a physical system to [Formula: see text] such that [Formula: see text] is the probability of the occurrence of an event when the system is in state [Formula: see text]. In particular, the question is answered whether a given (small) set of numerical events will give rise to the assumption that one deals with a classical physical system or a quantum-mechanical one.

Matrix periods and competition periods of Boolean Toeplitz matrices

In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring B={0,1}. Given subsets S and T of {1,…,n−1}, an n×n Toeplitz matrix A=Tn〈S;T〉 is defined to have 1 as the (i,j)-entry if and only if j−i∈S or i−j∈T. We show that if max⁡S+min⁡T≤n and min⁡S+max⁡T≤n, then A has the matrix period d/d′ and the competition period 1 where d=gcd⁡(s+t|s∈S,t∈T) and d′=gcd⁡(d,min⁡S). Moreover, it is shown that the limit of the matrix sequence {Am(AT)m}m=1∞ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.

Non-Trivial Subring Perfect Codes in Unit Graph of Boolean Rings

The aim of this paper is to investigate the non-trivial subring perfect codes in a unit graph associated with the Boolean rings. We prove a subring perfect code of size , where , in the unit graphs associated with the finite Boolean rings . Moreover, we give a necessary and sufficient condition for a subring of an infinite Boolean ring to be a perfect code of size infinity in the unit graph.

Conditional-completion of Boolean rings by lower cuts

Conditional-completion of Boolean rings by lower cuts

The prime ideals of the Boolean ring of intervals

The prime ideals of the Boolean ring of intervals

Commutativity of generalized Boolean rings

Commutativity of generalized Boolean rings

Generalizations of UU-rings, UJ-rings and UNJ-rings

A unit-picker is a map [Formula: see text] that associates to every ring [Formula: see text] a well-defined set [Formula: see text] of central units in [Formula: see text] which contains [Formula: see text], is invariant under isomorphisms of rings, is closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let [Formula: see text] be a unit-picker. A ring [Formula: see text] is called [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text]. An extensive study of these rings is conducted, and their connections with strongly nil [Formula: see text]-clean rings and semi [Formula: see text]-Boolean rings are investigated. When [Formula: see text] is specified, known results of [Formula: see text]-rings, [Formula: see text]-rings and [Formula: see text]-rings are obtained and new results are proved.

A Rickart‐Like Theorem for the Additivity of Multiplicative Maps on Rings

Rickart’s theorem states that every bijective multiplicative mapping of a Boolean ring R onto an arbitrary ring S is necessarily additive. We prove a version of Rickart’s theorem for non‐bijective mappings. This enables us to partially answer a question that was left open (Al Subaiei, B., Jarboui, N. On the Monoid of Unital Endomorphisms of a Boolean Ring. Axioms 2021, 10, 305).

On the Monoid of Unital Endomorphisms of a Boolean Ring

Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn.

Nil-Clean Rings with Involution

A [Formula: see text]-ring [Formula: see text] is called a nil [Formula: see text]-clean ring if every element of [Formula: see text] is a sum of a projection and a nilpotent. Nil [Formula: see text]-clean rings are the [Formula: see text]-version of nil-clean rings introduced by Diesl. This paper is about the nil [Formula: see text]-clean property of rings with emphasis on matrix rings. We show that a [Formula: see text]-ring [Formula: see text] is nil [Formula: see text]-clean if and only if [Formula: see text] is nil and [Formula: see text] is nil [Formula: see text]-clean. For a 2-primal [Formula: see text]-ring [Formula: see text], with the induced involution given by[Formula: see text], the nil [Formula: see text]-clean property of [Formula: see text] is completely reduced to that of [Formula: see text]. Consequently, [Formula: see text] is not a nil [Formula: see text]-clean ring for [Formula: see text], and [Formula: see text] is a nil [Formula: see text]-clean ring if and only if [Formula: see text] is nil, [Formula: see text]is a Boolean ring and [Formula: see text] for all [Formula: see text].

L -Fuzzy Prime Spectrums of ADLs

The notion of an Almost Distributive Lattice (ADL) is a common abstraction of several lattice theoretic and ring theoretic generalizations of Boolean algebra and Boolean rings. In this paper, the set of all L -fuzzy prime ideals of an ADL with truth values in a complete lattice L satisfying the infinite meet distributive law is topologized and the resulting space is discussed.