622 research outputs found
Representations of the Drazin inverse involving idempotents in a ring
We present some formulae for the Drazin inverse of difference and product of
idempotents in a ring. A number of results of bounded linear operators in
Banach spaces are extended to the ring case.Comment: 11 page
Drazin Invertibility of Product and Difference of Idempotents in a Ring
In this paper, several equivalent conditions on the Drazin invertibility of
product and difference of idempotents are obtained in a ring. Some results in
Banach algebra are extended to the ring case.Comment: 8 page
A Note on Quasi-Frobenius Rings
The Faith-Menal conjecture says that every strongly right ring is
. The conjecture is also equivalent to say every right noetherian left
-injective ring is . In this short article, we show that the conjecture
is true under the condition(a proper generalization of left condition)that
every nonzero complement left ideal is not small(a left ideal is called
small if for every left ideal , += implies =). It is also
proved that (1) is if and only if is a left and right mininjective
ring with on right annihilators in which ;
(2) is if and only if is a right simple injective ring with
on right annihilators in which . Several known
results on rings are obtained as corollaries.Comment: 8 page
Additive and product properties of Drazin inverses of elements in a ring
We study the Drazin inverses of the sum and product of two elements in a
ring. For Drazin invertible elements and such that and
, it is shown that is Drazin invertible and that is Drazin
invertible if and only if is Drazin invertible. Moreover, the formulae
of and are presented. Thus, a generalization of the main
result of Zhuang, Chen et al. (Linear Multilinear Algebra 60 (2012) 903-910) is
given
Weak group inverse
In this paper, we introduce a weak group inverse (called the WG inverse in
the present paper) for square matrices of an arbitrary index, and give some of
its characterizations and properties. Furthermore, we introduce two orders: one
is a pre-order and the other is a partial order, and derive several
characterizations of the two orders. At last, one characterization of the
core-EP order is derived by using the WG inverses.Comment: 22page
Additive property of pseudo Drazin inverse of elements in a Banach algebra
We study properties of pseudo Drazin inverse in a Banach algebra with unity
1. If and are pseudo Drazin invertible, we prove that is
pseudo Drazin invertible if and only if is pseudo Drazin
invertible. Moreover, the formula of is presented . When the
commutative condition is weaken to , we also
show that is pseudo Drazin invertible if and only if
is pseudo Drazin invertible
Strongly Goldie Dimension
Let be an associative ring with identity. A unital right -module
is called strongly finite dimensional if Sup. Properties of strongly finite dimensional modules are explored. It is
also proved that: (1)If is left -injective and strongly right finite
dimensional, then is left finite dimensional. (2) If is right
-injective, then is right finite dimensional if and only if is
semilocal. Thus the Faith-Menal conjecture is true if is strongly right
finite dimensional. Some known results are obtained as corollaries.Comment: 9 page
On countably -C2 rings
Let be a ring. is called a right countably -C2 ring if every
countable direct sum copies of is a C2 module. The following are
equivalent for a ring : (1) is a right countably -C2 ring. (2)
The column finite matrix ring
is a right C2 (or C3) ring. (3) Every countable direct sum copies of is
a C3 module. (4) Every projective right -module is a C2 (or C3) module. (5)
is a right perfect ring and every finite direct sum copies of is a
C2 (or C3) module. This shows that right countably -C2 rings are just
the rings whose right finitistic projective dimension
r=sup\{ is a right -module with
\}=0, which were introduced by Hyman Bass in 1960.Comment: 9 page
Small Injective Rings
Let be a ring, a right ideal of is called small if for every
proper right ideal of , . A ring is called right small
injective if every homomorphism from a small right ideal to can be
extended to an -homomorphism from to . Properties of small
injective rings are explored and several new characterizations are given for
rings and rings, respectively.Comment: 14 page
2-clean rings
A ring is said to be -clean if every element can be written as a sum
of an idempotent and units. The class of these rings contains clean ring
and -good rings in which each element is a sum of units. In this paper,
we show that for any ring , the endomorphism ring of a free -module of
rank at least 2 is 2-clean and that the ring of all row and column-finite matrices over any ring is 2-clean. Finally,
the group ring is considered where is a local ring. \vskip 0.5cm
{\bf Key words:}\quad 2-clean rings, 2-good rings, free modules, row and
column-finite matrix rings, group rings.Comment: 11 page
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