⁎-Clean rings; some clean and almost clean Baer ⁎-rings and von Neumann algebras

Vaš, Lia

doi:10.1016/j.jalgebra.2010.10.011

Cited by 35 publications

(44 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Clearly, * -clean rings are clean and strongly * -clean rings are strongly clean. It was shown in [7,11] that there exists a clean * -ring but not * -clean, and unit regular * -regular rings (which called * -unit regular rings in [7]) need not be strongly * -clean, which answered two questions raised by Vaš in [15].…”

Section: Introductionmentioning

confidence: 91%

“…In [15], Vaš asked whether there is an example of a * -ring that is clean but not * -clean. It was answered affirmatively in [7] and [11].…”

Section: * -Clean Ringsmentioning

confidence: 99%

“…A * -ring R is * -regular [2] if for every x in R there exists a projection p such that xR = pR. Following Vaš [15], an element of a * -ring R is (strongly) * -clean if it can be expressed as the sum of a unit and a projection (that commute), and R is (strongly) * -clean if all of its elements are (strongly) * -clean. Clearly, * -clean rings are clean and strongly * -clean rings are strongly clean.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Note on Strongly *-Clean Rings

Cui¹,

Wang²

2015

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. A * -ring R is called (strongly) * -clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of * -clean rings are considered. In particular, a new class of * -clean rings which called strongly π- * -regular are introduced. It is shown that R is strongly π- * -regular if and only if R is π-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of * -clean rings are discussed, and equivalent conditions among * -rings related to * -cleanness are obtained.

show abstract

Section: Introductionmentioning

confidence: 91%

“…In [15], Vaš asked whether there is an example of a * -ring that is clean but not * -clean. It was answered affirmatively in [7] and [11].…”

Section: * -Clean Ringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Note on Strongly *-Clean Rings

Cui¹,

Wang²

2015

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Recently the concept of strongly clean rings were considered for any * -ring. Vaš [12] calls a * -ring R strongly * -clean if each of its elements is the sum of a projection and a unit that commute with each other (see also [10]). …”

Section: Introductionmentioning

confidence: 99%

Strongly nil *-clean rings

Harmanci¹,

Chen²,

Özcan³

2017

Journal of Algebra Combinatorics Discrete Structures and Applications

View full text Add to dashboard Cite

A * -ring R is called strongly nil * -clean if every element of R is the sum of a projection and a nilpotent element that commute with each other. In this paper we investigate some properties of strongly nil * -rings and prove that R is a strongly nil * -clean ring if and only if every idempotent in R is a projection, R is periodic, and R/J(R) is Boolean. We also prove that a * -ring R is commutative, strongly nil * -clean and every primary ideal is maximal if and only if every element of R is a projection.

show abstract

“…Recently the concept of strongly clean rings are considered for any * -ring. Vaš [15] calls a * -ring R strongly * -clean if each of its elements is the sum of a projection and a unit that commute with each other (see also [14]). …”

Section: Introductionmentioning

confidence: 99%

On strongly nil clean rings

Chen

Sheibani

2016

Communications in Algebra

View full text Add to dashboard Cite

A * -ring R is called strongly nil * -clean if every element of R is the sum of a projection and a nilpotent element that commute with each other. In this paper we prove that R is a strongly nil * -clean ring if and only if every idempotent in R is a projection, R is periodic, and R/J(R) is Boolean. For any commutative * -ring R with µ * = µ, η * = η ∈ R, the algebraic extension R[i] = {a + bi | a, b ∈ R, i 2 = µi + η } is strongly nil * -clean if and only if R is strongly nil * -clean and µη is nilpotent. We also prove that a * -ring R is commutative, strongly nil * -clean and every primary ideal is maximal if and only if every element of R is a projection.

show abstract

⁎-Clean rings; some clean and almost clean Baer ⁎-rings and von Neumann algebras

Cited by 35 publications

References 24 publications

A Note on Strongly *-Clean Rings

A Note on Strongly *-Clean Rings

Strongly nil *-clean rings

On strongly nil clean rings

Contact Info

Product

Resources

About