Dinesh Khurana - Academia.edu (original) (raw)

Papers by Dinesh Khurana

Research paper thumbnail of Unit-Regularity of Regular Nilpotent Elements

arXiv (Cornell University), Sep 26, 2015

Let a be a regular element of a ring R. If either K := r R (a) has the exchange property or every... more Let a be a regular element of a ring R. If either K := r R (a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions R R = K ⊕ Xn ⊕ Yn = En ⊕ Xn ⊕ aYn, where Yn ⊆ a n R and En ∼ = R/aR. As applications we get easier proofs of the results that a strongly π-regular ring has stable range one and also that a strongly π-regular element whose every power is regular is unit-regular. An element a of a ring R is called strongly π-regular if both chains aR ⊇ a 2 R ⊇ a 3 R... and Ra ⊇ Ra 2 ⊇ Ra 3 ... stabilize. If every element of R is strongly π-regular, then R is called a strongly π-regular ring. In [1] Pere Ara proved a wonderful result that a strongly π-regular ring has stable range one. Ara's proof is on the following lines. As a strongly π-regular ring is an exchange ring and an exchange ring has stable range one if and only if every regular element is unit-regular, it is enough to show that every regular element of a strongly π-regular ring is unit-regular. Suppose a is a regular element of a strongly π-regular ring. By [7, Proposition 1] there exist n ∈ N, an idempotent e and a unit u in R with a n = eu such that a, e and u commute with each other. Then ea is a unit in eRe with inverse ea n−1 u −1 and (1 − e)a is a regular nilpotent element of the exchange ring (1 − e)R(1 − e). As a = ea + (1 − e)a and ea is unit-regular in eRe, we will get that a is unit-regular if we can show that (1 − e)a is unit-regular in (1 − e)R(1 − e). So the result will follow if we can show that a regular nilpotent element of an exchange ring is unit-regular. This is the crucial result proved by Ara in [1] and an easier proof of this will follow from our Theorem 2. Recently Ara and O'Meara in [2] and Pace andŠter in [8] have shown that a regular nilpotent element in general may not be unit-regular. By A ⊆ ⊕ B we shall mean that A is a summand of the module B. We will tacitly use the fact that a regular element a ∈ R is unit-regular if and only if r R (a) ∼ = R/aR, where r R (a) = {x ∈ R : ax = 0}. Lemma 1 [4, Corollary 3.9]. If M has the exchange property and A = M ⊕ B ⊕ C = I A i ⊕ C, then there exists a decomposition A i = D i ⊕ E i of each A i such that A = M ⊕ I D i ⊕ C.

Research paper thumbnail of A New Determinantal Formula for Three Matrices

Journal of Algebra and Its Applications

Research paper thumbnail of An ensemble of idempotent lifting hypotheses

Journal of Pure and Applied Algebra, Jun 1, 2018

Lifting idempotents modulo ideals is an important tool in studying the structure of rings. This p... more Lifting idempotents modulo ideals is an important tool in studying the structure of rings. This paper lays out the consequences of lifting other properties modulo ideals, including lifting of von Neumann regular elements, lifting isomorphic idempotents, and lifting conjugate idempotents. Applications are given for IC rings, perspective rings, and Dedekind-finite rings, which improve multiple results in the literature. We give a new characterization of the class of exchange rings; they are rings where regular elements lift modulo all left ideals. We also uncover some hidden connections between these lifting properties. For instance, if regular elements lift modulo an ideal, then so do isomorphic idempotents. The converse is true when units lift. The logical relationships between these and several other important lifting properties are completely characterized. Along the way, multiple examples are developed that illustrate limitations to the theory.

Research paper thumbnail of Invertible Commutators in Matrix Rings

Journal of Algebra and Its Applications, Feb 1, 2011

In a matrix ring R = 𝕄2(S) where S is a commutative ring, we study equations of the form XY - YX ... more In a matrix ring R = 𝕄2(S) where S is a commutative ring, we study equations of the form XY - YX = U ∈ GL 2(S), focusing on matrices in R that can appear as X or as XY in such equations. These are the completable and the reflectable matrices in R. For matrices A ∈ R with a zero row or with a constant diagonal, explicit and "computer-checkable" criteria are found for A to be completable or reflectable. A formula for det (XY - YX) discovered recently with Shomron connects this study to diophantine questions about the representation of units of the ground ring S by quadratic forms of the type px2 + qy2.

Research paper thumbnail of Endomorphism Rings of Finite Length Injective Modules

Kyungpook Mathematical Journal, Jun 1, 2003

We prove the existence of a finite length injective right module whose endomorphism ring is not r... more We prove the existence of a finite length injective right module whose endomorphism ring is not right Artinian.

Research paper thumbnail of Noncommutative rings, group rings, diagram algebras and their applications : international conference, December 18-22, 2006, University of Madras, Chennai, India

American Mathematical Society eBooks, 2008

... Representation Theory and Statistical Mechanics P. MARTIN 99 Robinson-Schensted Correspondenc... more ... Representation Theory and Statistical Mechanics P. MARTIN 99 Robinson-Schensted Correspondence for the G-Brauer Algebras M. PARVATHI AND A ... We also wish to express our appreciation to the Patrons Dr. S. Ramachandran Vice Chancellor, University of Madras and ...

Research paper thumbnail of Unit-Regularity of Regular Nilpotent Elements

Algebras and Representation Theory, Jan 5, 2016

Let a be a regular element of a ring R. If either K := r R (a) has the exchange property or every... more Let a be a regular element of a ring R. If either K := r R (a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions R R = K ⊕ Xn ⊕ Yn = En ⊕ Xn ⊕ aYn, where Yn ⊆ a n R and En ∼ = R/aR. As applications we get easier proofs of the results that a strongly π-regular ring has stable range one and also that a strongly π-regular element whose every power is regular is unit-regular. An element a of a ring R is called strongly π-regular if both chains aR ⊇ a 2 R ⊇ a 3 R... and Ra ⊇ Ra 2 ⊇ Ra 3 ... stabilize. If every element of R is strongly π-regular, then R is called a strongly π-regular ring. In [1] Pere Ara proved a wonderful result that a strongly π-regular ring has stable range one. Ara's proof is on the following lines. As a strongly π-regular ring is an exchange ring and an exchange ring has stable range one if and only if every regular element is unit-regular, it is enough to show that every regular element of a strongly π-regular ring is unit-regular. Suppose a is a regular element of a strongly π-regular ring. By [7, Proposition 1] there exist n ∈ N, an idempotent e and a unit u in R with a n = eu such that a, e and u commute with each other. Then ea is a unit in eRe with inverse ea n−1 u −1 and (1 − e)a is a regular nilpotent element of the exchange ring (1 − e)R(1 − e). As a = ea + (1 − e)a and ea is unit-regular in eRe, we will get that a is unit-regular if we can show that (1 − e)a is unit-regular in (1 − e)R(1 − e). So the result will follow if we can show that a regular nilpotent element of an exchange ring is unit-regular. This is the crucial result proved by Ara in [1] and an easier proof of this will follow from our Theorem 2. Recently Ara and O'Meara in [2] and Pace andŠter in [8] have shown that a regular nilpotent element in general may not be unit-regular. By A ⊆ ⊕ B we shall mean that A is a summand of the module B. We will tacitly use the fact that a regular element a ∈ R is unit-regular if and only if r R (a) ∼ = R/aR, where r R (a) = {x ∈ R : ax = 0}. Lemma 1 [4, Corollary 3.9]. If M has the exchange property and A = M ⊕ B ⊕ C = I A i ⊕ C, then there exists a decomposition A i = D i ⊕ E i of each A i such that A = M ⊕ I D i ⊕ C.

Research paper thumbnail of Transitivity of Perspectivity

Algebras and Representation Theory, Jan 25, 2021

We study modules in which perspectivity of summands is transitive. Generalizing a 1977 result of ... more We study modules in which perspectivity of summands is transitive. Generalizing a 1977 result of Handelman and a 2014 result of Garg, Grover, and Khurana, we prove that for any ring R, perspectivity is transitive in M 2 (R) if and only if R has stable range one. Also generalizing a 2019 result of Amini, Amini, and Momtahan we prove that a quasi-continuous module in which perspectivity is transitive is perspective.

Research paper thumbnail of On a Question of Hartwig and Luh

Bulletin of The Australian Mathematical Society, Jun 13, 2013

In 1977 Hartwig and Luh asked whether an element a in a Dedekind-finite ring R satisfying aR = a ... more In 1977 Hartwig and Luh asked whether an element a in a Dedekind-finite ring R satisfying aR = a 2 R also satisfies Ra = Ra 2. In this paper, we answer this question in the negative. We also prove that if a is an element of a Dedekind-finite exchange ring R and aR = a 2 R, then Ra = Ra 2. This gives an easier proof of Dischinger's theorem that left strongly π-regular rings are right strongly π-regular, when it is already known that R is an exchange ring.

Research paper thumbnail of Unit sum numbers of right self-injective rings

Bulletin of The Australian Mathematical Society, Jun 1, 2007

Research paper thumbnail of Right Self-Injective Rings in Which Every Element is a Sum of Two Units

Journal of Algebra and Its Applications, Apr 1, 2007

A classical result of Zelinsky states that every linear transformation on a vector space V , exce... more A classical result of Zelinsky states that every linear transformation on a vector space V , except when V is one-dimensional over Z 2 , is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if no factor ring of R is isomorphic to Z 2 .

Research paper thumbnail of Semiperfect rings and Nakayama permutations

Glasgow Mathematical Journal, May 1, 2002

We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its bas... more We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its basic idempotents. We also give a few necessary and sufficient conditions for a semiperfect ring R, which cogenerates every 2-generated right R-module, to be right pseudo-Frobenius.

Research paper thumbnail of Lifting potent elements modulo nil ideals

Journal of Pure and Applied Algebra, Nov 1, 2021

Abstract It is classically known that idempotents lift modulo nil one-sided ideals. So it is natu... more Abstract It is classically known that idempotents lift modulo nil one-sided ideals. So it is natural to ask if the same is true for potent elements. Although we answer this question in negative, we prove that the answer is positive in several special cases. For instance, the answer is positive in rings with finite characteristic. Let I be a nil one sided ideal of R and x ∈ R is such that x n + 1 − x ∈ I for some positive integer n. We prove that if n is a unit in R, then there exists an element p ∈ R with p n + 1 = p such that x − p ∈ I . By taking n = 1 , the result that idempotents lift modulo nil one sided ideals is retrieved. For a nil ideal I of an abelian ring R, we prove that potent elements lift modulo I precisely when torsion units or periodic elements lift modulo I. It follows that torsion units or periodic elements may also not lift modulo nil ideals. We prove that torsion units, potent elements and periodic elements lift modulo every nil ideal of a π-regular ring.

Research paper thumbnail of Endomorphism rings projective covers and lifting of idempotents

Research paper thumbnail of A short proof of Grinshpon's theorem

arXiv (Cornell University), Apr 6, 2009

Research paper thumbnail of On GCD and LCM in domains — A conjecture of gauss

Research paper thumbnail of Group Rings that are Additively Generated by Idempotents and Units

arXiv (Cornell University), Apr 6, 2009

Let R be an Abelian 1 exchange ring. We prove the following results: 1. RZ 2 and RS 3 are clean r... more Let R be an Abelian 1 exchange ring. We prove the following results: 1. RZ 2 and RS 3 are clean rings. 2. If G is a group of prime order p and p is in the Jacobson radical of R, then RG is clean. 3. If identity in R is a sum of two units and G is a locally finite group, then every element in RG is a sum of two units. 4. For any locally finite group G, RG has stable range one. All rings in this note are associative with identity. An element of a ring is said to be clean if it is a sum of a unit and an idempotent. A ring R is said to be clean if its every element is clean. These rings were introduced by Nicholson in [N 1 ] as a class of examples of exchange rings. In [N 1 , Proposition 1.8] Nicholson proved that an Abelian exchange ring is clean. This work is motivated by the paper [M] of McGovern where it is proved that for a commutative clean ring R, the group ring RZ 2 is clean. We extend this result by proving that RZ 2 is clean whenever R is an Abelian exchange ring. Moreover our proof is quite short. We also prove that RS 3 is clean for any Abelian exchange ring R.

Research paper thumbnail of A Characterization of Unit Regular Rings

Communications in Algebra, Apr 30, 2001

Research paper thumbnail of A note on completeness and strongly clean rings

Journal of Pure and Applied Algebra, Apr 1, 2014

Many authors have investigated the behavior of strong cleanness under certain ring extensions. In... more Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we investigate the classical problem of lifting idempotents, in order to consolidate and extend these results. Our main result is that if R is a ring which is complete with respect to an ideal I and if x is an element of R whose image in R/I is strongly π-regular, then x is strongly clean in R. This generalizes Theorem 2.1 of Chen and Zhou (2007) [9].

Research paper thumbnail of Some Characterizations of VNL Rings

arXiv (Cornell University), Jan 16, 2008

A ring R is said to be VNL if for any a ∈ R, either a or 1−a is (von Neumann) regular. The class ... more A ring R is said to be VNL if for any a ∈ R, either a or 1−a is (von Neumann) regular. The class of VNL rings lies properly between the exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without infinite set of orthogonal idempotents; and also the VNL rings having primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a 1 , a 2) ∈ R 2 , one of the a i is regular in R. Formal triangular matrix rings that are VNL, are also characterized. As a corollary it is shown that an upper triangular matrix ring T n (R) is VNL if and only if n = 2 or 3 and R is a division ring.

Research paper thumbnail of Unit-Regularity of Regular Nilpotent Elements

arXiv (Cornell University), Sep 26, 2015

Let a be a regular element of a ring R. If either K := r R (a) has the exchange property or every... more Let a be a regular element of a ring R. If either K := r R (a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions R R = K ⊕ Xn ⊕ Yn = En ⊕ Xn ⊕ aYn, where Yn ⊆ a n R and En ∼ = R/aR. As applications we get easier proofs of the results that a strongly π-regular ring has stable range one and also that a strongly π-regular element whose every power is regular is unit-regular. An element a of a ring R is called strongly π-regular if both chains aR ⊇ a 2 R ⊇ a 3 R... and Ra ⊇ Ra 2 ⊇ Ra 3 ... stabilize. If every element of R is strongly π-regular, then R is called a strongly π-regular ring. In [1] Pere Ara proved a wonderful result that a strongly π-regular ring has stable range one. Ara's proof is on the following lines. As a strongly π-regular ring is an exchange ring and an exchange ring has stable range one if and only if every regular element is unit-regular, it is enough to show that every regular element of a strongly π-regular ring is unit-regular. Suppose a is a regular element of a strongly π-regular ring. By [7, Proposition 1] there exist n ∈ N, an idempotent e and a unit u in R with a n = eu such that a, e and u commute with each other. Then ea is a unit in eRe with inverse ea n−1 u −1 and (1 − e)a is a regular nilpotent element of the exchange ring (1 − e)R(1 − e). As a = ea + (1 − e)a and ea is unit-regular in eRe, we will get that a is unit-regular if we can show that (1 − e)a is unit-regular in (1 − e)R(1 − e). So the result will follow if we can show that a regular nilpotent element of an exchange ring is unit-regular. This is the crucial result proved by Ara in [1] and an easier proof of this will follow from our Theorem 2. Recently Ara and O'Meara in [2] and Pace andŠter in [8] have shown that a regular nilpotent element in general may not be unit-regular. By A ⊆ ⊕ B we shall mean that A is a summand of the module B. We will tacitly use the fact that a regular element a ∈ R is unit-regular if and only if r R (a) ∼ = R/aR, where r R (a) = {x ∈ R : ax = 0}. Lemma 1 [4, Corollary 3.9]. If M has the exchange property and A = M ⊕ B ⊕ C = I A i ⊕ C, then there exists a decomposition A i = D i ⊕ E i of each A i such that A = M ⊕ I D i ⊕ C.

Research paper thumbnail of A New Determinantal Formula for Three Matrices

Journal of Algebra and Its Applications

Research paper thumbnail of An ensemble of idempotent lifting hypotheses

Journal of Pure and Applied Algebra, Jun 1, 2018

Lifting idempotents modulo ideals is an important tool in studying the structure of rings. This p... more Lifting idempotents modulo ideals is an important tool in studying the structure of rings. This paper lays out the consequences of lifting other properties modulo ideals, including lifting of von Neumann regular elements, lifting isomorphic idempotents, and lifting conjugate idempotents. Applications are given for IC rings, perspective rings, and Dedekind-finite rings, which improve multiple results in the literature. We give a new characterization of the class of exchange rings; they are rings where regular elements lift modulo all left ideals. We also uncover some hidden connections between these lifting properties. For instance, if regular elements lift modulo an ideal, then so do isomorphic idempotents. The converse is true when units lift. The logical relationships between these and several other important lifting properties are completely characterized. Along the way, multiple examples are developed that illustrate limitations to the theory.

Research paper thumbnail of Invertible Commutators in Matrix Rings

Journal of Algebra and Its Applications, Feb 1, 2011

In a matrix ring R = 𝕄2(S) where S is a commutative ring, we study equations of the form XY - YX ... more In a matrix ring R = 𝕄2(S) where S is a commutative ring, we study equations of the form XY - YX = U ∈ GL 2(S), focusing on matrices in R that can appear as X or as XY in such equations. These are the completable and the reflectable matrices in R. For matrices A ∈ R with a zero row or with a constant diagonal, explicit and "computer-checkable" criteria are found for A to be completable or reflectable. A formula for det (XY - YX) discovered recently with Shomron connects this study to diophantine questions about the representation of units of the ground ring S by quadratic forms of the type px2 + qy2.

Research paper thumbnail of Endomorphism Rings of Finite Length Injective Modules

Kyungpook Mathematical Journal, Jun 1, 2003

We prove the existence of a finite length injective right module whose endomorphism ring is not r... more We prove the existence of a finite length injective right module whose endomorphism ring is not right Artinian.

Research paper thumbnail of Noncommutative rings, group rings, diagram algebras and their applications : international conference, December 18-22, 2006, University of Madras, Chennai, India

American Mathematical Society eBooks, 2008

... Representation Theory and Statistical Mechanics P. MARTIN 99 Robinson-Schensted Correspondenc... more ... Representation Theory and Statistical Mechanics P. MARTIN 99 Robinson-Schensted Correspondence for the G-Brauer Algebras M. PARVATHI AND A ... We also wish to express our appreciation to the Patrons Dr. S. Ramachandran Vice Chancellor, University of Madras and ...

Research paper thumbnail of Unit-Regularity of Regular Nilpotent Elements

Algebras and Representation Theory, Jan 5, 2016

Let a be a regular element of a ring R. If either K := r R (a) has the exchange property or every... more Let a be a regular element of a ring R. If either K := r R (a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions R R = K ⊕ Xn ⊕ Yn = En ⊕ Xn ⊕ aYn, where Yn ⊆ a n R and En ∼ = R/aR. As applications we get easier proofs of the results that a strongly π-regular ring has stable range one and also that a strongly π-regular element whose every power is regular is unit-regular. An element a of a ring R is called strongly π-regular if both chains aR ⊇ a 2 R ⊇ a 3 R... and Ra ⊇ Ra 2 ⊇ Ra 3 ... stabilize. If every element of R is strongly π-regular, then R is called a strongly π-regular ring. In [1] Pere Ara proved a wonderful result that a strongly π-regular ring has stable range one. Ara's proof is on the following lines. As a strongly π-regular ring is an exchange ring and an exchange ring has stable range one if and only if every regular element is unit-regular, it is enough to show that every regular element of a strongly π-regular ring is unit-regular. Suppose a is a regular element of a strongly π-regular ring. By [7, Proposition 1] there exist n ∈ N, an idempotent e and a unit u in R with a n = eu such that a, e and u commute with each other. Then ea is a unit in eRe with inverse ea n−1 u −1 and (1 − e)a is a regular nilpotent element of the exchange ring (1 − e)R(1 − e). As a = ea + (1 − e)a and ea is unit-regular in eRe, we will get that a is unit-regular if we can show that (1 − e)a is unit-regular in (1 − e)R(1 − e). So the result will follow if we can show that a regular nilpotent element of an exchange ring is unit-regular. This is the crucial result proved by Ara in [1] and an easier proof of this will follow from our Theorem 2. Recently Ara and O'Meara in [2] and Pace andŠter in [8] have shown that a regular nilpotent element in general may not be unit-regular. By A ⊆ ⊕ B we shall mean that A is a summand of the module B. We will tacitly use the fact that a regular element a ∈ R is unit-regular if and only if r R (a) ∼ = R/aR, where r R (a) = {x ∈ R : ax = 0}. Lemma 1 [4, Corollary 3.9]. If M has the exchange property and A = M ⊕ B ⊕ C = I A i ⊕ C, then there exists a decomposition A i = D i ⊕ E i of each A i such that A = M ⊕ I D i ⊕ C.

Research paper thumbnail of Transitivity of Perspectivity

Algebras and Representation Theory, Jan 25, 2021

We study modules in which perspectivity of summands is transitive. Generalizing a 1977 result of ... more We study modules in which perspectivity of summands is transitive. Generalizing a 1977 result of Handelman and a 2014 result of Garg, Grover, and Khurana, we prove that for any ring R, perspectivity is transitive in M 2 (R) if and only if R has stable range one. Also generalizing a 2019 result of Amini, Amini, and Momtahan we prove that a quasi-continuous module in which perspectivity is transitive is perspective.

Research paper thumbnail of On a Question of Hartwig and Luh

Bulletin of The Australian Mathematical Society, Jun 13, 2013

In 1977 Hartwig and Luh asked whether an element a in a Dedekind-finite ring R satisfying aR = a ... more In 1977 Hartwig and Luh asked whether an element a in a Dedekind-finite ring R satisfying aR = a 2 R also satisfies Ra = Ra 2. In this paper, we answer this question in the negative. We also prove that if a is an element of a Dedekind-finite exchange ring R and aR = a 2 R, then Ra = Ra 2. This gives an easier proof of Dischinger's theorem that left strongly π-regular rings are right strongly π-regular, when it is already known that R is an exchange ring.

Research paper thumbnail of Unit sum numbers of right self-injective rings

Bulletin of The Australian Mathematical Society, Jun 1, 2007

Research paper thumbnail of Right Self-Injective Rings in Which Every Element is a Sum of Two Units

Journal of Algebra and Its Applications, Apr 1, 2007

A classical result of Zelinsky states that every linear transformation on a vector space V , exce... more A classical result of Zelinsky states that every linear transformation on a vector space V , except when V is one-dimensional over Z 2 , is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if no factor ring of R is isomorphic to Z 2 .

Research paper thumbnail of Semiperfect rings and Nakayama permutations

Glasgow Mathematical Journal, May 1, 2002

We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its bas... more We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its basic idempotents. We also give a few necessary and sufficient conditions for a semiperfect ring R, which cogenerates every 2-generated right R-module, to be right pseudo-Frobenius.

Research paper thumbnail of Lifting potent elements modulo nil ideals

Journal of Pure and Applied Algebra, Nov 1, 2021

Abstract It is classically known that idempotents lift modulo nil one-sided ideals. So it is natu... more Abstract It is classically known that idempotents lift modulo nil one-sided ideals. So it is natural to ask if the same is true for potent elements. Although we answer this question in negative, we prove that the answer is positive in several special cases. For instance, the answer is positive in rings with finite characteristic. Let I be a nil one sided ideal of R and x ∈ R is such that x n + 1 − x ∈ I for some positive integer n. We prove that if n is a unit in R, then there exists an element p ∈ R with p n + 1 = p such that x − p ∈ I . By taking n = 1 , the result that idempotents lift modulo nil one sided ideals is retrieved. For a nil ideal I of an abelian ring R, we prove that potent elements lift modulo I precisely when torsion units or periodic elements lift modulo I. It follows that torsion units or periodic elements may also not lift modulo nil ideals. We prove that torsion units, potent elements and periodic elements lift modulo every nil ideal of a π-regular ring.

Research paper thumbnail of Endomorphism rings projective covers and lifting of idempotents

Research paper thumbnail of A short proof of Grinshpon's theorem

arXiv (Cornell University), Apr 6, 2009

Research paper thumbnail of On GCD and LCM in domains — A conjecture of gauss

Research paper thumbnail of Group Rings that are Additively Generated by Idempotents and Units

arXiv (Cornell University), Apr 6, 2009

Let R be an Abelian 1 exchange ring. We prove the following results: 1. RZ 2 and RS 3 are clean r... more Let R be an Abelian 1 exchange ring. We prove the following results: 1. RZ 2 and RS 3 are clean rings. 2. If G is a group of prime order p and p is in the Jacobson radical of R, then RG is clean. 3. If identity in R is a sum of two units and G is a locally finite group, then every element in RG is a sum of two units. 4. For any locally finite group G, RG has stable range one. All rings in this note are associative with identity. An element of a ring is said to be clean if it is a sum of a unit and an idempotent. A ring R is said to be clean if its every element is clean. These rings were introduced by Nicholson in [N 1 ] as a class of examples of exchange rings. In [N 1 , Proposition 1.8] Nicholson proved that an Abelian exchange ring is clean. This work is motivated by the paper [M] of McGovern where it is proved that for a commutative clean ring R, the group ring RZ 2 is clean. We extend this result by proving that RZ 2 is clean whenever R is an Abelian exchange ring. Moreover our proof is quite short. We also prove that RS 3 is clean for any Abelian exchange ring R.

Research paper thumbnail of A Characterization of Unit Regular Rings

Communications in Algebra, Apr 30, 2001

Research paper thumbnail of A note on completeness and strongly clean rings

Journal of Pure and Applied Algebra, Apr 1, 2014

Many authors have investigated the behavior of strong cleanness under certain ring extensions. In... more Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we investigate the classical problem of lifting idempotents, in order to consolidate and extend these results. Our main result is that if R is a ring which is complete with respect to an ideal I and if x is an element of R whose image in R/I is strongly π-regular, then x is strongly clean in R. This generalizes Theorem 2.1 of Chen and Zhou (2007) [9].

Research paper thumbnail of Some Characterizations of VNL Rings

arXiv (Cornell University), Jan 16, 2008

A ring R is said to be VNL if for any a ∈ R, either a or 1−a is (von Neumann) regular. The class ... more A ring R is said to be VNL if for any a ∈ R, either a or 1−a is (von Neumann) regular. The class of VNL rings lies properly between the exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without infinite set of orthogonal idempotents; and also the VNL rings having primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a 1 , a 2) ∈ R 2 , one of the a i is regular in R. Formal triangular matrix rings that are VNL, are also characterized. As a corollary it is shown that an upper triangular matrix ring T n (R) is VNL if and only if n = 2 or 3 and R is a division ring.