Boolean Rings Research Articles

A Note on Boolean Like Algebras

In this paper we develop on abstract system: viz Boolean-like algebra and prove that every Boolean algebra is a Boolean-like algebra. A necessary and sufficient condition for a Boolean-like algebra to be a Boolean algebra has been obtained. As in the case of Boolean ring and Boolean algebra, it is established that under suitable binary operations the Boolean-like ring and Boolean-like algebra are equivalent abstract structures.

Some Contributions to Boolean like near Rings

In this paper we extend Foster’s Boolean-like ring to Near-rings. We introduce the concept of a Boolean like near-ring. A near-ring N is said to be a Boolean-like near-ring if the following conditions hold: (i) a+a = 0 for all aÎ N , (ii) ab(a+b+ab) = ba for all a, b Î N and (iii) abc = acb for all a,b, c Î N (right weak commutative law). We have proved that every Boolean ring is a Boolean like near-ring. An example is given to show that the converse is not true. We prove that if N is a Boolean near-ring then conditions (i) and (ii) of the above definition are equivalent. We also proved that a Boolean near-ring with condition (iii) is a Boolean ring. As a consequence we show that a Boolean –like near-ring N is a Boolean ring if and only if it is a Boolean near-ring. Obviously, every Boolean like ring is a Boolean like near-ring. We show that if N is a Boolean-like near-ring with identity, then N is a Boolean-like ring. In addition we prove several interesting properties of Boolean-like near-rings. We prove that the set of all nilpotent elements of a Boolean –like near-ring N forms an ideal and the quotient near-ring N/I is a Boolean ring. Every homomorphic image of a Boolean like near ring is a Boolean like near ring. We further prove that every Boolean-like near-ring is a Boolean-like semiring As example is given to show that the converse of this result is not true.

Ideals and their complements in commutative semirings

We study conditions under which the lattice {{mathrm{mathbf {Id}}}}mathbf R of ideals of a given a commutative semiring {mathbf {R}} is complemented. At first we check when the annihilator I^* of a given ideal I of {mathbf {R}} is a complement of I. Further, we study complements of annihilator ideals. Next we investigate so-called Łukasiewicz semirings. These form a counterpart to MV-algebras which are used in quantum structures as they form an algebraic semantic of many-valued logics as well as of the logic of quantum mechanics. We describe ideals and congruence kernels of these semirings with involution. Finally, using finite unitary Boolean rings, a construction of commutative semirings with complemented lattice of ideals is presented.

Structure of Zhou Nil-clean Rings

A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ϵ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM(R) is 5-potent and BM(R) is nil, if and only if J(R) is nil and R/J(R) is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn(R; s) is Zhou nil-clean if and only if R is Zhou nil-clean and s ϵ J(R).

English

A *-ring R is strongly 2-nil-*-clean if every element in R is the sum of two projections and a nilpotent that commute. Fundamental properties of such *-rings are obtained. We prove that a *-ring R is strongly 2-nil-*-clean if and only if for all a ∈ R, a2 ∈ R is strongly nil-*-clean, if and only if for any a ∈ R there exists a *-tripotent e ∈ R such that a − e ∈ R is nilpotent and ea = ae, if and only if R is a strongly *-clean SN ring, if and only if R is abelian, J(R) is nil and R/J(R) is *-tripotent. Furthermore, we explore the structure of such rings and prove that a *-ring R is strongly 2-nil-*-clean if and only if R is abelian and R ≅ R1,R2 or R1 × R2, where R1/J(R1) is a *-Boolean ring and J(R1) is nil, R2/J(R2) is a *-Yaqub ring and J(R2) is nil. The uniqueness of projections of such rings are thereby investigated.

Reduction graph and its application on algebraic graphs

Evans, et al. (J. Graph Theory 18 (1994), 801--815), defined an equivalence relation $\sim $ on the set of vertices of a simple graph $G$ by taking $a\sim b$ if and only if their open neighborhoods are the same. They introduced a new graph $G_{red }={G}/{\sim } $, reduction graph of $G$, as follows. The vertices are $V (G_{red })=\{[a]: a \in V (G) \}$, and two equivalence classes $[a]$ and $[b]$ are adjacent if and only if $a$ and $b$ are adjacent in $G$. Recently, Anderson and LaGrange defined some equivalence relations on the set of vertices of the zero-divisor graph of a commutative ring, one of which yields the reduction graph of the zero-divisor graph. In this paper, we state some basic graph theoretic properties of $G_{red }$ and study the relations between some properties of graph $G$ and its subgraph, $G_{red }$, such as the chromatic number, clique number, girth and diameter. Moreover, we study the reduction graph of some algebraic graphs, such as the comaximal graph, zero-divisor graph and Cayley graph of a commutative ring. Among other results, we show that, for every commutative ring $R$, $\Gamma _2(R)_{red } \simeq \Gamma _1(\mathbb {Z}_2^n)$, where $\Gamma _1(\mathbb {Z}_2^n)$ is the zero-divisor graph of the Boolean ring $\mathbb {Z}_2^n$, $\Gamma _2(R)$ is the comaximal graph of $R$ and $n=|Max (R)|$.

Local dimension theory of tensor products of algebras over a ring

Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring [Formula: see text]. Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field [Formula: see text] into the general setting of algebras over an arbitrary ring [Formula: see text]. For this sake, we introduce and study the notion of a fibered AF-ring over a ring [Formula: see text]. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc. 19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings [Formula: see text] consisting of two [Formula: see text]-algebras [Formula: see text] and [Formula: see text] such that [Formula: see text], we introduce the inherent notion to [Formula: see text] of a [Formula: see text]-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product [Formula: see text]. As an application, we provide a formula for the Krull dimension of [Formula: see text] when either [Formula: see text] or [Formula: see text] is zero-dimensional as well as for the Krull dimension of [Formula: see text] when [Formula: see text] is a fibered AF-ring over the ring of integers [Formula: see text] with nonzero characteristic and [Formula: see text] is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of [Formula: see text] when [Formula: see text] is a Boolean ring. Actually, we prove that if [Formula: see text] and [Formula: see text] are rings such that [Formula: see text] is not trivial and [Formula: see text] is a Boolean ring, then dim[Formula: see text].

L-Fuzzy Filters of Almost Distributive Lattices

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 ring. A complete lattice satisfying the infinite meet distributive law is called a frame. In this paper, we introduce the notion of $L$-fuzzy filter of a given ADL $A$ with a maximal element, where $L$ is a given frame and here we apply the general theory of algebraic fuzzy systems to the class of $L-$fuzzy filters of an ADL.

English

We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; Z3 ⊕ Z3; Z3 ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M2(Z2) or M2(Z3); or the ring of a Morita context with zero pairings where the underlying rings are Z2 or Z3.

The idempotent-divisor graphs of a commutative ring

ABSTRACTGiven an idempotent element e of a commutative ring R, define the idempotent-divisor graph Γe(R) of R associated with e to be the (undirected) graph with vertex-set V(Γe(R)) = {a∈R | there exists b∈R with ab = e} such that distinct a,b∈V(Γe(R)) are adjacent if and only if ab = e. In this paper, the interplay between the algebraic properties of R and the graph-theoretic structure of Γe(R) is investigated. For example, it is shown that if Re is a Boolean ring then Γe(R) is connected, and if R has identity then Γe(R) is finite if and only if R(1−e) and U(R) are finite, where U(R) is the group of units of R. Furthermore, if e∈R and Γe(T(R)) is connected then , where T(R) is the total quotient ring of R. Emphasis is given to linking the structure of Γe(R) with that of zero-divisor graphs (i.e., idempotent-divisor graphs associated with 0).

When does a generalized Boolean quasiring become a Boolean ring?

Generalized Boolean quasirings are ring-like structures used as algebraic models in the foundations of axiomatic quantum mechanics. The quantum mechanical system corresponding to such a quasiring turns out to be a classical one if and only if this quasiring is a Boolean ring with unit. We characterize this situation by a single identity.

POINTWISE CONVERGENCE TOPOLOGY AND FUNCTION SPACES IN FUZZY ANALYSIS

We study the space of all continuous fuzzy-valued functions from a space $X$ into the space of fuzzy numbers $(mathbb{E}sp{1},dsb{infty})$ endowed with the pointwise convergence topology. Our results generalize the classical ones for continuous real-valued functions. The field of applications of this approach seems to be large, since the classical case allows many known devices to be fitted to general topology, functional analysis, coding theory, Boolean rings, etc.

Boolean Differential Calculus

Abstract The Boolean Differential Calculus (BDC) is a very powerful theory that extends the basic concepts of Boolean Algebras significantly. Its applications are based on Boolean spaces 𝔹 and 𝔹ⁿ, Boolean operations, and basic structures such as Boolean Algebras and Boolean Rings, Boolean functions, Boolean equations, Boolean inequalities, incompletely specified Boolean functions, and Boolean lattices of Boolean functions. These basics, sometimes also called switching theory, are widely used in many modern information processing applications. The BDC extends the known concepts and allows the consideration of changes of function values. Such changes can be explored for pairs of function values as well as for whole subspaces. The BDC defines a small number of derivative and differential operations. Many existing theorems are very welcome and allow new insights due to possible transformations of problems. The available operations of the BDC have been efficiently implemented in several software packages. The ...

Quantum one-way permutation over the finite field of two elements

In quantum cryptography, a one-way permutation is a bounded unitary operator $U:\mathcal{H} \to \mathcal{H}$ on a Hilbert space $\mathcal{H}$ that is easy to compute on every input, but hard to invert given the image of a random input. Levin [Probl. Inf. Transm., vol. 39 (1): 92-103 (2003)] has conjectured that the unitary transformation $g(a,x)=(a,f(x)+ax)$, where $f$ is any length-preserving function and $a,x \in GF_{{2}^{\|x\|}}$, is an information-theoretically secure operator within a polynomial factor. Here, we show that Levin's one-way permutation is provably secure because its output values are four maximally entangled two-qubit states, and whose probability of factoring them approaches zero faster than the multiplicative inverse of any positive polynomial $poly(x)$ over the Boolean ring of all subsets of $x$. Our results demonstrate through well-known theorems that existence of classical one-way functions implies existence of a universal quantum one-way permutation that cannot be inverted in subexponential time in the worst ca

English

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ℕ. We prove that M n (R) is nil clean if and only if R/J(R) is Boolean and M n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ℤ3, B or ℤ3 ⊕ B where B is a Boolean ring, and that M n (R) is weakly nil clean if and only if M n (R) is nil clean for all n ≥ 2.

Boolean Rings Obtained from Hyperrings with $$ \boldsymbol{\eta}_{1,m}^{\boldsymbol{*}} $$ η 1 , m ∗ Relations

In this paper, we introduce and analyze the smallest equivalence binary relation \( \eta_{1,m}^{*} \) on a hyperring R such that the quotient \( R/\eta_{1,m}^{*}\), the set of all equivalence classes, is a commutative ring with identity and for any x ∊ R, \( [\eta_{1,m}^{*} (x)]^{m + 1} = \eta_{1,m}^{*} (x) \). The characterization of Boolean rings via strongly regular relations is investigated, and some properties on the topic are presented.

Transient dynamics and their control in time-delay autonomous Boolean ring networks.

Biochemical systems with switch-like interactions, such as gene regulatory networks, are well modeled by autonomous Boolean networks. Specifically, the topology and logic of gene interactions can be described by systems of continuous piecewise-linear differential equations, enabling analytical predictions of the dynamics of specific networks. However, most models do not account for time delays along links associated with spatial transport, mRNA transcription, and translation. To address this issue, we have developed an experimental test bed to realize a time-delay autonomous Boolean network with three inhibitory nodes, known as a repressilator, and use it to study the dynamics that arise as time delays along the links vary. We observe various nearly periodic oscillatory transient patterns with extremely long lifetime, which emerge in small network motifs due to the delay, and which are distinct from the eventual asymptotically stable periodic attractors. For repeated experiments with a given network, we find that stochastic processes give rise to a broad distribution of transient times with an exponential tail. In some cases, the transients are so long that it is doubtful the attractors will ever be approached in a biological system that has a finite lifetime. To counteract the long transients, we show experimentally that small, occasional perturbations applied to the time delays can force the trajectories to rapidly approach the attractors.

Relative Pseudo-Complementations on ADL's

The notion of an Almost Distributive Lattice (abbreviated as ADL) is a common abstraction of several lattice theoretic generalization of Boolean algebras and Boolean rings. In this paper we introduce the notion of Relative pseudo-complementation on ADL's and discuss several properties of this.

Additive unit structure of endomorphism rings and invariance of modules

We use the type theory for rings of operators due to Kaplansky to describe the structure of modules that are invariant under automorphisms of their injective envelopes. Also, we highlight the importance of Boolean rings in the study of such modules. As a consequence of this approach, we are able to further the study initiated by Dickson and Fuller regarding when a module invariant under automorphisms of its injective envelope is invariant under any endomorphism of it. In particular, we find conditions for several classes of noetherian rings which ensure that modules invariant under automorphisms of their injective envelopes are quasi-injective. In the case of a commutative noetherian ring, we show that any automorphism-invariant module is quasi-injective. We also provide multiple examples to show that our conditions are the best possible, in the sense that if we relax them further then there exist automorphism-invariant modules which are not quasi-injective. We finish this paper by dualizing our results to the automorphism-coinvariant case.

On (m, n)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a1⋯am∈I for a1,…, am∈R∖U(R), then there are n of the ai’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bezout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m − 1 maximal ideals.