Vertex and region colorings of planar idempotent divisor graphs of commutative rings.

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ϵ R, and is denoted by Л(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number, the chromatic number and the region chromatic number for every planar idempotent divisor graphs of commutative rings. Also we show the clique number is equal to the chromatic number for any planar idempotent divisor graph. Among other results we prove that: Let Fq, Fpa are fieldes of orders q and pa respectively, where q=2 or 3, p is a prime number and a Is a positive integer. If a ring R @ Fq x Fpa . Then (Л(R))= (Л(R)) = *( Л(R)) = 3.

Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}). In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

On generalized ∗-reverse derivable maps

Let R be a ring with involution containing a nontrivial symmetric idempotent element e and δ: R → R be a generalized ∗-reverse derivable map. In this paper, our aim is to show that under some suitable restrictions imposed on R every generalized ∗-reverse derivable map of R is additive.

A note on r-precious ring

An element of a ring [Formula: see text] is said to be [Formula: see text]-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.

Resolving resolution dimension of recollements of abelian categories

In this paper, we investigate the resolving resolution dimension with respect to the recollements of abelian categories. Let [Formula: see text] be a recollement of abelian categories such that [Formula: see text] and [Formula: see text] have enough projective objects and let [Formula: see text], [Formula: see text], [Formula: see text] be resolving subcategories of [Formula: see text], [Formula: see text] and [Formula: see text], respectively. We establish some upper and lower bounds of [Formula: see text]-resolution dimension of [Formula: see text] in terms of the [Formula: see text]-resolution dimension of [Formula: see text] and [Formula: see text]-resolution dimension of [Formula: see text]. Based on these upper and lower bounds, we study the Gorensteinness of abelian categories involved in [Formula: see text]. Under some suitable assumptions, we show that if [Formula: see text] and [Formula: see text] are Gorenstein, then [Formula: see text] is Gorenstein. As applications, we apply our results to ring theory and the triangular matrix artin algebras, we study the quasi-Frobenius and Gorenstein hereditary properties of the ring [Formula: see text] and [Formula: see text], where [Formula: see text] is an idempotent element of [Formula: see text]. We also investigate Gorensteinness of the triangular matrix artin algebras, some known results are obtained as corollaries. At the end of this paper, we give two examples to illustrate our results.

Clean graph of a ring

Let [Formula: see text] be a ring (not necessarily commutative) with identity. The clean graph [Formula: see text] of a ring [Formula: see text] is a graph with vertices in form [Formula: see text], where [Formula: see text] is an idempotent and [Formula: see text] is a unit of [Formula: see text]; and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we focus on [Formula: see text], the subgraph of [Formula: see text] induced by the set [Formula: see text] is a nonzero idempotent element of [Formula: see text]. It is observed that [Formula: see text] has a crucial role in [Formula: see text]. The clique number, the chromatic number, the independence number and the domination number of the clean graph for some classes of rings are determined. Moreover, the connectedness and the diameter of [Formula: see text] are studied.

Some results on $$\mathcal {L}$$-commutative semigroups

We prove first that every $$\mathcal {H}$$ H -commutative semigroup is stable. Using this result [and some results from the standard text (Nagy, Special classes of semigroups, Kluwer, Dordrecht, 2001)], we give two equivalent conditions for a semigroup to be an archimedean $$\mathcal {H}$$ H -commutative semigroup containing an idempotent element. It turns out that this result can be partially extended to $$\mathcal {L}$$ L -commutative semigroups and quasi-commutative semigroups.

GENERALIZED JORDAN DERIVATIONS ON SEMIPRIME RINGS

The purpose of this note is to prove the following. Suppose $\mathfrak{R}$ is a semiprime unity ring having an idempotent element $e$ ($e\neq 0,~e\neq 1$) which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on $\mathfrak{R}$ is a generalized derivation.

Antiautomorphisms with quasi-generalized Engel condition

Let [Formula: see text] be a ring with 1. Given elements [Formula: see text], [Formula: see text] of [Formula: see text] and the integer [Formula: see text] define [Formula: see text] and [Formula: see text]. We say that a given antiautomorphism [Formula: see text] of [Formula: see text] is commuting if [Formula: see text], all [Formula: see text]. More generally, assume that [Formula: see text] satisfies the condition [Formula: see text] where [Formula: see text], [Formula: see text] are corresponding positive integers depending on [Formula: see text], and [Formula: see text] ranges over [Formula: see text]. To what extent can one say that [Formula: see text] is commuting? In this paper, we answer the question in the affirmative if R is a prime ring containing some idempotent element [Formula: see text]. In the diametrically opposed case in which [Formula: see text] is a division ring the answer is again yes provided [Formula: see text] is algebraic over its center and [Formula: see text] is of finite order. These two major complementary results will be put to work to provide an answer to the general question.

ON THE SET-THEORETIC STRENGTH OF ELLIS’ THEOREM AND THE EXISTENCE OF FREE IDEMPOTENT ULTRAFILTERS ON ω

Ellis’ Theorem (i.e., “every compact Hausdorff right topological semigroup has an idempotent element”) is known to be proved only under the assumption of the full Axiom of Choice (AC); AC is used in the proof in the disguise of Zorn’s Lemma.In this article, we prove that in ZF, Ellis’ Theorem follows from the Boolean Prime Ideal Theorem (BPI), and hence is strictly weaker than AC in ZF. In fact, we establish that BPI implies the formally stronger (than Ellis’ Theorem) statement “for every family ${\cal A} = \{ ({S_i},{ \cdot _i},{{\cal T}_i}):i \in I\}$ of nontrivial compact Hausdorff right topological semigroups, there exists a function f with domain I such that $f\left( i \right)$ is an idempotent of ${S_i}$, for all $i \in I$”, which in turn implies ACfin (i.e., AC for sets of nonempty finite sets).Furthermore, we prove that in ZFA, the Axiom of Multiple Choice (MC) implies Ellis’ Theorem for abelian semigroups (i.e., “every compact Hausdorff right topological abelian semigroup has an idempotent element”) and that the strictly weaker than MC (in ZFA) principle LW (i.e., “every linearly ordered set can be well-ordered”) implies Ellis’ Theorem for linearly orderable semigroups (i.e., “every compact Hausdorff right topological linearly orderable semigroup has an idempotent element”); thus the latter formally weaker versions of Ellis’ Theorem are strictly weaker than BPI in ZFA. Yet, it is shown that no choice is required in order to prove Ellis’ Theorem for well-orderable semigroups.We also show that each one of the (strictly weaker than AC) statements “the Tychonoff product $2^{\Cal R} $ is compact and Loeb” and $BPI_{\Cal R}$ (BPI for filters on ${\Cal R}$) implies “there exists a free idempotent ultrafilter on ω” (which in turn is not provable in ZF). Moreover, we prove that the latter statement does not imply $BP{I_\omega }$ (BPI for filters on ω) in ZF, hence it does not imply any of $AC_{\Cal R} $ (AC for sets of nonempty sets of reals) and $BPI_{\Cal R} $ in ZF, either.In addition, we prove that the statements “there exists a free ultrafilter on ω”, “there exists a free ultrafilter on ω which is not idempotent”, and “for every IP set $A \subseteq \omega$, there exists a free ultrafilter ${\cal F}$ on ω such that $A \in {\cal F}$” are pairwise equivalent in ZF.