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Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

For a finite Coxeter group W and a Coxeter element c of W ; the c -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W . Its maximal cones are naturally indexed by the c -sortable elements of W . The main result of this paper is that the known bijection cl c between c -sortable elements and c -clusters induces a combinatorial isomorphism of fans. In particular, the c -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W . The rays...

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.

Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring

Szigeti, Jeno (2006)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0, where λ0+λ1x+…+λn-1 xn-1+n!xn is the right characteristic polynomial of A in R[x], I ∈ Mn(R) is the identity matrix and the entries of the n×n matrices Ci, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then C0 = C1 = … = Cn-1 = Cn = 0 and our...

Cellular covers of cotorsion-free modules

Rüdiger Göbel, José L. Rodríguez, Lutz Strüngmann (2012)

Fundamenta Mathematicae

In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism π ⁎ : H o m R ( G , G ) ≅ H o m R ( G , H ) , where π⁎(φ) = πφ for each φ ∈ H o m R ( G , G ) (where maps are acting on the left). On the one hand,...

Central Armendariz rings.

Agayev, Nazim, Güngöroğlu, Gonca, Harmanci, Abdullah, Halicioğlu, S. (2011)

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

Certain decompositions of matrices over Abelian rings

Nahid Ashrafi, Marjan Sheibani, Huanyin Chen (2017)

Czechoslovak Mathematical Journal

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ℕ . We prove that M n ( R ) is nil clean if and only if R / J ( R ) is Boolean and M n ( J ( R ) ) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R / J ( R ) is ℤ 3 , B or ℤ 3 ⊕ B where B is a Boolean ring, and that M n ( R ) is weakly nil clean if and only if M n ( R ) is nil clean for all n ≥ 2 .

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