Let A be a unital complex semisimple Banach algebra, and MA denote its maximal ideal space. For matrix M∈An×n, Mˆ denotes the obtained by taking entry-wise Gelfand transforms. M∈Cn×n, σ(M)⊂C set of eigenvalues M. It is shown that if A∈An×n B∈Am×m are such for all φ∈MA, σ(Aˆ(φ))∩σ(Bˆ(φ))=∅, then C∈An×m, Sylvester equation AX−XB=C has unique solution X∈An×m. As an application, Roth's removal rule...

Let A be the ring obtained by localizing the polynomial ring κ[X, Y, Z, W ] over a field κ at the maximal ideal (X, Y, Z, W) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext i A (M, A/p), where M is a module construct...

Let A be the ring obtained by localizing the polynomial ring κ[X,Y, Z,W ] over a field κ at the maximal ideal (X, Y, Z,W ) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y . We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext A (M,A/p), where M is a module constructed b...

We prove the following result on the universal localization of a ring R at an ideal I: If the universal localization is flat as an R-module, then R satisfies the Ore condition with respect to the multiplicative set of elements that become invertible modulo I. It is well known that for domains the converse of this result holds, and hence we have found in this case a new characterization of the O...

Let C be a unital AH-algebra and let A be a unital separable simple C∗-algebra with tracial rank zero. Suppose that φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(a)ut = φ2(a) for all a ∈ C if and only if [φ1] = [φ2] in KK(C,A), τ ◦ φ1 = τ ◦ φ2 for all τ ∈ T (A) and the rotation map η̃φ1,φ2 associate...

We study applications of discrete valuations to ideals in analytically irreducible domains, in particular applications to zero divisors modulo powers of ideals. We prove a uniform version of Izumi's theorem and calculate several examples illustrating it, such as for rational singularities. The paper contains a new criterion of analytic irreducibility, a new criterion of one-beredness, and a val...

We investigate the complexity of partial order relation Young's lattice. The definable relations are characterized by establishing maximal definability property modulo single automorphism given conjugation; consequently, as an ordered set lattice has undecidable elementary theory and is non-finitely axiomatizable but every ideal generates a finitely universal class equivalence relations. show p...

We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between ideals and microlocal V-filtration modulo Jacobian ideal. Via Tjurina subspectrum, we can compare with Steenbrink which be defined by V-filtration. As a consequence of formula Mustață Popa, these two spectra coincide in weighted homogeneous case. prove sufficient conditions their coincidence ...