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An infinite torus braid yields a categorified Jones-Wenzl projector

Lev Rozansky (2014)

Fundamenta Mathematicae

A sequence of Temperley-Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones-Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may serve as a categorification of the Jones-Wenzl projector.

Bases of minimal elements of some partially ordered free abelian groups

Pavel Příhoda (2003)

Commentationes Mathematicae Universitatis Carolinae

In the present paper, we will show that the set of minimal elements of a full affine semigroup A 0 k contains a free basis of the group generated by A in k . This will be applied to the study of the group K 0 ( R ) for a semilocal ring R .

Categorifications of the polynomial ring

Mikhail Khovanov, Radmila Sazdanovic (2015)

Fundamenta Mathematicae

We develop a diagrammatic categorification of the polynomial ring ℤ[x]. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to xⁿ and standard modules to (x-1)ⁿ in the Grothendieck ring.

Diagonal reductions of matrices over exchange ideals

Huanyin Chen (2006)

Czechoslovak Mathematical Journal

In this paper, we introduce related comparability for exchange ideals. Let I be an exchange ideal of a ring R . If I satisfies related comparability, then for any regular matrix A M n ( I ) , there exist left invertible U 1 , U 2 M n ( R ) and right invertible V 1 , V 2 M n ( R ) such that U 1 V 1 A U 2 V 2 = diag ( e 1 , , e n ) for idempotents e 1 , , e n I .

Exchange rings in which all regular elements are one-sided unit-regular

Huanyin Chen (2008)

Czechoslovak Mathematical Journal

Let R be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in R is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.

Exchange rings satisfying the related comparability.

Huanyin Chen, Fu-An Li (2002)

Collectanea Mathematica

In this paper we investigate the related comparability over exchange rings. It is shown that an exchange ring R satisfies the related comparability if and only if for any regular x C R, there exists a related unit w C R and a group G in R such that wx C G.

Exchange rings with stable range one

Huanyin Chen (2007)

Czechoslovak Mathematical Journal

We characterize exchange rings having stable range one. An exchange ring R has stable range one if and only if for any regular a R , there exist an e E ( R ) and a u U ( R ) such that a = e + u and a R e R = 0 if and only if for any regular a R , there exist e r . a n n ( a + ) and u U ( R ) such that a = e + u if and only if for any a , b R , R / a R R / b R a R b R .

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