Extensions of Clean Rings

Han, Juncheol; Nicholson, W. K.

doi:10.1081/agb-100002409

Cited by 166 publications

(112 citation statements)

References 8 publications

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“…(i) Example 3.3 leads to the result in [1] and [16] which says that a finite direct product of commutative rings is clean iff each of those rings is clean.…”

Section: Now Let Us Definementioning

confidence: 99%

“…Idempotent Lifting Property (ILP) is studied in ring theory ( [1], [4], [16], [31]), while Boolean Lifting Property (BLP) appears in MV-algebras ( [10]), BL-algebras ( [22]) and residuated lattices ([12], [13]). In this section we define a general notion of lifting property in the context of universal algebras; this notion of lifting property embodies ILP, BLP and many other important kinds of lifting properties.…”

Section: Lifting Properties In Universal Algebrasmentioning

confidence: 99%

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Algebraic and topological results on lifting properties in residuated lattices

Georgescu

Cheptea

Mureșan

2015

Fuzzy Sets and Systems

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We define lifting properties for universal algebras, which we study in this general context and then particularize to various such properties in certain classes of algebras. Next we focus on residuated lattices, in which we investigate lifting properties for Boolean and idempotent elements modulo arbitrary, as well as specific kinds of filters. We give topological characterizations to the lifting property for Boolean elements and several properties related to it, many of which we obtain by means of the reticulation.

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“…(i) Example 3.3 leads to the result in [1] and [16] which says that a finite direct product of commutative rings is clean iff each of those rings is clean.…”

Section: Now Let Us Definementioning

confidence: 99%

Section: Lifting Properties In Universal Algebrasmentioning

confidence: 99%

Algebraic and topological results on lifting properties in residuated lattices

Georgescu

Cheptea

Mureșan

2015

Fuzzy Sets and Systems

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“…Then R/J(R) is exchange and idempotents can be lifted modulo J(R) by [26, Proposition 1.4] and Abelian, and so R/J(R) is clean and idempotents lift modulo J(R). Thus R is clean by [13,Proposition 6].…”

Section: Clearly R Is Feckly Armendariz If and Only If Whenevermentioning

confidence: 99%

On Jacobson and Nil Radicals Related to Polynomial Rings

Kwak¹,

Lee²,

Özcan³

2016

Journal of the Korean Mathematical Society

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Abstract. This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that J(R[x]) = J(R) [x] if and only if J(R) is nil when a given ring R is Armendariz, where J(A) means the Jacobson radical of a ring A. A ring will be called feckly Armendariz if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

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“…If 2 ∈ U (R), then RC 2 ∼ = R ⊕ R by [9,Proposition 3]. Since R is a quasipolar ring, so is R ⊕ R by (1).…”

mentioning

confidence: 99%

Quasipolar Matrix Rings Over Local Rings

Cui¹,

Yin²

2014

Bulletin of the Korean Mathematical Society

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Abstract. A ring R is called quasipolar if for every a ∈ R there exists p 2 = p ∈ R such that p ∈ comm 2 R (a), a + p ∈ U (R) and ap ∈ R qnil . The class of quasipolar rings lies properly between the class of strongly π-regular rings and the class of strongly clean rings. In this paper, we determine when a 2 × 2 matrix over a local ring is quasipolar. Necessary and sufficient conditions for a 2 × 2 matrix ring to be quasipolar are obtained.

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Extensions of Clean Rings

Cited by 166 publications

References 8 publications

Algebraic and topological results on lifting properties in residuated lattices

Algebraic and topological results on lifting properties in residuated lattices

On Jacobson and Nil Radicals Related to Polynomial Rings

Quasipolar Matrix Rings Over Local Rings

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