New characterizations for core inverses in rings with involution

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakic, N. C. Dincic and D. S. Djordjevc generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a,b ∈ R. Then a ∈ R with a= b if and only if （ab）^＊ = ab, ba^2 = a, and ab^2 = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.

On Characterizations of Special Elements in Rings with Involution

Let R be a ring with involution. It is well-known that an EP element in R is a core invertible element, but the question when a core invertible element is an EP element,the authors answer in this paper. Several new characterizations of star-core, normal and Hermitian elements in R are also presented.

Approximations and cotorsion pairs related to a tilting pair

在 artin 代数学上的倾斜的对的观点被 Miyashita 在 2001 介绍。它是在倾斜的理论的一个有用工具。与固定的倾斜的对有关的近似和 cotorsion 对被讨论。一 contravariantly (covariantly ) 有限子范畴和与固定的倾斜的对联系的 cotorsion 对在这份报纸被给。

Weighted binary relations involving core-EP inverse

We study a new binary relation defined on the set of rectangular complex matrices involving the weighted core-EP inverse and give its characterizations.This relation becomes a pre-order.Then,one-sided preorders associated to the weighted core-EP inverse are given from two perspectives.Finally,we make a comparison for these two sets of one-sided weighted pre-orders.

Characterizations for the Core Invertibility and EP-ness Involving Projections

In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of 1−pq,1−pqp,p−pq and p−q,where p,q are projections in different settings,such as∗-rings,∗-reducing rings and C∗-algebras.Moreover,several representations for the core inverses of product,difference and sum of two generalized projections are derived.In particular,a number of examples are given to illustrate our results.

On Some n-C2 and Strongly C2 Extensions

Let R be a ring and n be a positive integer.Then R is called a left n-C2-ring(strongly left C2-ring)if every n-generated(finitely generated)proper right ideal of R has nonzero left annihilator.We discuss some n-C2 and strongly C2 extensions,such as trivial extensions,formal triangular matrix rings,group rings and[D,C].

The Pseudo Drazin inverses in Banach Algebras

Let A be a complex Banach algebra and J be the Jacobson radical of A.(1)We firstly show that a is generalized Drazin invertible in A if and only if a+J is generalized Drazin invertible in A/J.Then we prove that a is pseudo Drazin invertible in si if and only if a+J is Drazin invertible in A/J.As its application,the pseudo Drazin invertibility of elements in a Banach algebra is explored.(2)The pseudo Drazin order is introduced in A.We give the necessary and sufficient conditions under which elements in A have pseudo Drazin order,then we prove that the pseudo Drazin order is a pre-order.

Some ＊-Clean Group Ringst

A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that is not ＊-clean. In this paper, we investigate when a group ring RG is ＊-clean, where ＊ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ＊-clean, where R is a commutative local ring and G is one of the groups C3,C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ＊-clean.