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.

Quasi-clean rings and strongly quasi-clean rings

An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see text] is called a quasi-Boolean ring if every element of [Formula: see text] is quasi-idempotent. A ring [Formula: see text] is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or [Formula: see text] has no image isomorphic to [Formula: see text]; For an indecomposable commutative semilocal ring [Formula: see text] with at least two maximal ideals, [Formula: see text]([Formula: see text]) is strongly quasi-clean if and only if [Formula: see text] is quasi-clean if and only if [Formula: see text], [Formula: see text] is a maximal ideal of [Formula: see text]. For a prime [Formula: see text] and a positive integer [Formula: see text], [Formula: see text] is strongly quasi-clean if and only if [Formula: see text]. Some open questions are also posed.

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].

Spectra of Boolean Graphs Over Finite Fields of Characteristic Two

With entries of the adjacency matrix of a simple graph being regarded as elements of $\mathbb{F}_{2}$, it is proved that a finite commutative ring $R$ with $1\neq 0$ is a Boolean ring if and only if either $R\in \{\mathbb{F}_{2},\mathbb{F}_{2}\times \mathbb{F}_{2}\}$ or the eigenvalues (in the algebraic closure of $\mathbb{F}_{2}$) corresponding to the zero-divisor graph of $R$ are precisely the elements of $\mathbb{F}_{4}\setminus \{0\}$ . This is achieved by observing a way in which algebraic behavior in a Boolean ring is encoded within Pascal’s triangle so that computations can be carried out by appealing to classical results from number theory.

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.

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).

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].