Representations of the Drazin inverse involving idempotents in a ring

An element a of a ring R is called Drazin invertible if there exists b∈R such that ab =ba,bab =b,and a -a2 b is nilpotent.The element b above is unique if it exists and is denoted as aD .The equivalent conditions of the Drazin inverse involving idempotents in R are established.As applications, some formulae for the Drazin inverse of the difference and the product of idempotents in a ring are given.Hence,a number of results of bounded linear operators in Banach spaces are extended to the ring case.

Using involutory and idempotent matrices over finite fields to construct cartesian authentication codes

Two constructions of Cartesian authentication codes form involutory and idempotent matrices over finitefields are presented and their size Parameters are computed. Moreover, assume that the encoding rules are chosen ac-cording to a uniform Probablity distribution, the PI and Ps which dend the largest Probablities of a successful impersonation attack and a successful substitution attack respectively of these codes are also computed. Finially, it is Provedthat thase two Cartesian authentication codes are isomorphic.

ON THE INVERTIBILITY OF HILBERT SPACE IDEMPOTENTS

This note is to present some results on the group invertibility of linear combina- tions of idempotents when the difference of two idempotents is group invertible.

Generalized n-idempotents and Hyper-generalized n-idempotents

For an integer n ≥2, we say that an operator A is an n-idempotent if A^n = A; A is a generalized n-idempotent if A^n = A^＊; A is a hyper-generalized n. idempotent if A^n = A^＋. The set of all n-idempotents, all generalized n-idempotents and all hyper-generalized n-idempotents are denoted by In（H）, gIn（H） and HgIn（H）, respectively. In this note, we obtain a chain of proper inclusions gIn（H） belong to HgIn（H） belong to In＋2（H）.

Using Normal Form of Idempotent Matrices over Finite Local Ring Z/p^kZ to Construct Cartesian Authentication Codes

In this paper, we determine the normal forms of idempotent matrices for similarity over finite local rings Z/p^kZ, from which we construct a Cartesian authentication code and compute its size parameters and the probabilities of successful impersonation and substitution attack under the hypothesis that the cecoding rules are chosen according to a uniform probability distribution.

RIESZ IDEMPOTENT OF (n, k)-QUASI-*-PARANORMAL OPERATORS

A bounded linear operator T on a complex Hilbert space H is called（n, k）-quasi-*-paranormal if ║T;（T;x） ║;║ T;x║;≥║ T*（T;x）║ for all x ∈ H,where n, k are nonnegative integers. This class of operators has many interesting properties and contains the classes of n-*-paranormal operators and quasi-*-paranormal operators. The aim of this note is to show that every Riesz idempotent E;with respect to a non-zero isolated spectral point λ of an（n, k）-quasi-*-paranormal operator T is self-adjoint and satisfies ran E;= ker（T- λ） = ker（T- λ）*.

THE EXISTENCE OF OVERLARGE SETS OF IDEMPOTENT QUASIGROUPS

A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n+1) into n+1 n(n-1)×3 partial orthogonal arrays A_x, x∈X based on X\{x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.

On Primitive Idempotents of the Strong Endomorphism Monoid of a Graph

In this paper, we describe the canonical partial order on the idempotent set of the strong endomorphism monoid of a graph, and using this we further characterize primitive idem potenes from the viewpoint of combinatorics. The number of them is also given.

A CLASS OF NATURALLY ORDERED ABUNDANT SEMIGROUPS WITH A GREATESTIDEMPOTENT

On the base of the construction of abundant semigroups with a normal medial idempotent [14], in this paper we consider a class of naturally ordered abundant semigroups which satisfies the regularity condition and contains a greatest idempotent. Furthermore, we give a completely description of the overall structure of such ordered semigroups via the algebraic structure of them, which generalizes known result obtained by Blyth and McFadden[3].

Idempotent Elements of the Semigroups BX(D) Defined by Semilattices of the Class ∑3 (X,8) When Z7=Ø

In this paper, complete semigroup binary relation is defined by semilattices of the class . We give a full description of idempotent elements of given semigroup. For the case where X is a finite set and , we derive formulas by calculating the numbers of idempotent elements of the respective semigroup.

Idempotent Elements of the Semigroups Bx(D) Defined by Semilattices of the Class ∑3(x,8) When Z7‡Ø

In the paper, complete semigroup binary relation is defined by semilattices of the class . We give a full description of idempotent elements of given semigroup. For the case where X is a finite set and , we derive formulas by calculating the numbers of idempotent elements of the respective semigroup.

On the Number of Idempotent Partial Contraction Mappings of a Finite Chain

Let be the partial symmetric semigroup on and let and be its subsemigroups of order-preserving contractions and order-preserving, order-decreasing contractions mappings of , respectively. In this paper we investigate the cardinalities of and , the set idempotents of and , respectively. We also investigate the cardinalities of certain equivalences on and .

On Diagonalization of Idempotent Matrices over APT Rings

Let R be an abelian ring (all idempotents of R lie in the center of R), and A be an idempotent matrix over R. The following statements are proved: (a). A is equivalent to a diagonal matrix if and only if A is similar to a diagonal matrix. (b). If R is an APT (abelian projectively trivial) ring, then A can be uniquely diagonalized as diag{el, ..., en} and ei divides ei+1. (c). R is an APT ring if and only if R/I is an APT ring, where I is a nilpotent ideal of R. By (a), we prove that a separative abelian regular ring is an APT ring.

The lattice of idempotent distributive semiring varieties

A solution is given for the word problem for free idempotent distributive semirings. Using this solution the lattice L(ID) of subvarieties of the variety ID of idempotent distributive semirings is determined. It turns out that L(ID) is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the lattice L (B) of all band varieties. Therefore L(ID) is countably infinite and distributive. Every subvariety of ID is finitely based.

Skew Pairs of Idempotents in Transformation Semigroups

An ordered pair （e, f） of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. We have shown previously that there are four distinct types of skew pairs of idempotents. Here we investigate the ubiquity of such skew pairs in full transformation semigroups.

FULLY IDEMPOTENT RINGS WHOSE EVERY MAXIMAL LEFT IDEAL IS AN IDEAL

In [1] the following problem was proposed: Is a ring R strongly regular if every ideal of R is idempotent and every maximal left ideal of R is an ideal? In [2], it was asked whether a ring R is von Neumann regular if every ideal of R is idempotent and every maximal essential left ideal of R is an ideal? The primary aim of this note is to construct a counterexample for the above questions.

Some Rank Equalities about Combinations of Two Idempotent Matrices

The paper researches the rank of combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.Using the properties of the idempotent matrix and elementary block matrix operation,we get some rank equalities for combinations a PA＋bAQ-cPAQ of two idempotent matrices P and Q.These rank equalities generalize the results of Koliha J J,Rakoevi V and Tian Y,and give some applications of the rank equalities.

Homological behavior of idempotent subalgebras and Ext algebras

Let A be a(left and right) Noetherian ring that is semiperfect. Let e be an idempotent of A and consider the ring Γ :=(1-e)A(1-e) and the semi-simple right A-module Se := e A/e rad A. In this paper, we investigate the relationship between the global dimensions of A and Γ, by using the homological properties of Se. More precisely, we consider the Yoneda ring Y(e) := Ext_A~*(Se, Se) of e. We prove that if Y(e) is Artinian of finite global dimension, then A has finite global dimension if and only if so does Γ. We also investigate the situation where both A and Γ have finite global dimension. When A is Koszul and finite dimensional, this implies that Y(e) has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if Y(e) has finite global dimension, then the Cartan determinants of A and Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.

Idempotent Completions of Operator Partial Matrices

Necessary and sufficient conditions are obtained for operator partial 2×2 matrices to have an idempotent completion, and all such completions are parametrically represented.

Orthogonality of the idempotent elements with rank one in primitive rings

Let R be a primitive ring with nonzero socle, M a faithful irreducible right R-module, A the central-izer of M, and L= a direct sum of countably many minimal right ideals L, of R. Then there existsa family of subsetsis infinite) of R such that L=R for any W, whereeach is a set of countably many orthogonal idempotent elements of rank one in R. Furthermore,there exists a primitive ring R and a direct sum L=of countably many minimal right ideals Li ofR, but R has no subset B =of countably many orthogonal idempotent elements of rank one such that and B can be extended to a corresponding basis of some basis of M over A.