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An element $a$ in a ring $R$ has a ps-Drazin inverse if there exists $b\in comm^2(a)$ such that $b=bab, (a-ab)^k\in J(R)$ for some $k\in {\Bbb N}$. Elementary properties of ps-Drazin inverses in a ring are investigated here. We prove that $a\in R$ has a ps-Drazin inverse if and only if $a$ has a generalized Drazin inverse and $(a-a^2)^k\in J(R)$ for some $k\in {\Bbb N}$. We show Cline's formula and Jacobson's lemma for ps-Drazin inverses. The additive properties of ps-Drazin inverses in a Banach algebra are obtained. Moreover, we completely determine when a $2\times 2$ matrix $A\in M_2(R)$ over a local ring $R$ has a ps-Drazin inverse.

Let R be a local ring and suppose that there exists a is an element of F* such that a(6) not equal 1; also let T : M-n(R) -> M-m(R) be a linear map preserving Drazin inverses. Then we prove that T = 0 or n = m and T preserves idempotents. We thereby determine the form of linear maps from M-n(R) to M-m(R) preserving Drazin inverses of matrices.

Let R be a ring with identity, M be a right R-module and F be a fully invariant submodule of M. The concept of an F-inverse split module M has been investigated recently. In this paper, we approach to this concept with a different perspective, that is, we deal with a notion of an F-image split module M, and study various properties and obtain some characterizations of this kind of modules. By means of F-image split modules M, we focus on modules M in which fully invariant submodules F are dual Rickart direct summands. In this way, we contribute to the notion of a T-dual Rickart module M by considering Z?2 (M) as the fully invariant submodule F of M. We also deal with a notion of relatively image splitness to investigate direct sums of image split modules. Some applications of image split modules to rings are given.

We are concerned about various strongly clean properties of a kind of matrix subrings $L_{(s)}(R)$ over a local ring $R$. Let $R$ be a local ring, and let $s\in C(R)$. We prove that $A\in L_{(s)}(R)$ is strongly clean if and only if $A$ or $I_2-A$ is invertible, or $A$ is similar to a diagonal matrix in $L_{(s)}(R)$. Furthermore, we prove that $A\in L_{(s)}(R)$ is quasipolar if and only if $A\in GL_2(R)$ or $A\in L_{(s)}(R)^{qnil}$, or $A$ is similar to a diagonal matrix $\left( \begin{array}{cc} \lambda&0\\ 0&\mu \end{array} \right)$ in $L_{(s)}(R)$, where $\lambda\in J(R)$, $\mu\in U(R)$ or $\lambda\in U(R)$, $\mu\in J(R)$, and $l_{\mu}-r_{\lambda}$, $l_{\lambda}-r_{\mu}$ are injective. Pseudopolarity of such matrix subrings is also obtained.

In this paper, we introduce a concept of a dual F-Baer module M where F is the fully invariant submodule of M, by this means we deal with generating dual Baer modules. We investigate direct sums of dual F-Baer modules M by exerting the notion of relatively dual F-Baer modules. We also obtain applications of dual F-Baer modules to rings and the preradical Z * (.).Mathematics Subject Classification (2010): 16D10, 16D40, 16D70, 16D80.Keywords: Dual Baer module, dual inverse split module, fully invariant submodule, cosingular submodule

In this paper, we deal with a new approach to quasipolarity notion for rings, namely an element a of a ring R is called weakly nil-quasipolar if there exists \(p^2 = p\in comm^2(a)\) such that \(a + p\) or \(a-p\) is nilpotent, and the ring R is called weakly nil-quasipolar if every element of R is weakly nil-quasipolar. The class of weakly nil-quasipolar rings lies properly between the classes of nil-quasipolar rings and quasipolar rings. Although it is an open problem whether strongly clean (even quasipolar) rings have stable range one, we show that there is an affirmative answer for weakly nil-quasipolar rings. It is also proved that if R is a weakly nil-quasipolar NI ring, then R / N(R) is commutative. Moreover, we consider the question of when certain \(2 \times 2\) matrices over a commutative local ring is weakly nil-quasipolar.

Let R be an arbitrary ring with identity and M a right R-module with S= End(R)(M). Let F be a fully invariant submodule of M and I-1(F) denotes the set {m is an element of M : Im subset of F} for any subset I of S. The module M is called F-Baer if I-1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F circle plus N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.

In this paper, we introduce a new notion which lies properly between strong pi-regularity and pseudopolarity. A ring R is feckly polar if for any a is an element of R there exists e is an element of comm(2) (a) such that a - e is an element of U(R), e - e(2) is an element of J(R) and (ae)(n) is an element of J(R) for some n is an element of N. Many structure theorems are proved. Further, we investigate feck polarity for triangular matrix and matrix rings. The relations among strongly pi-regular rings, pseudopolar rings and feckly polar rings are also obtained.

In this paper, we introduce a class of quasipolar rings which is a generalization of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a \in R$ is called {\it $\delta$-quasipolar} if there exists $p^2 = p\in comm^2(a)$ such that $a + p$ is contained in $\delta(R)$, and the ring $R$ is called {\it $\delta$-quasipolar} if every element of $R$ is $\delta$-quasipolar. We use $\delta$-quasipolar rings to extend some results of $J$-quasipolar rings. Then some of the main results of $J$-quasipolar rings are special cases of our results for this general setting. We give many characterizations and investigate general properties of $\delta$-quasipolar rings., Comment: Submitted for publication