we call a module  essentially retractable if homr for all essential submodules n of m. for a right fbn ring r, it is shown that: (i)  a non-zero module  is retractable (in the sense that homr for all non-zero ) if and only if certain factor modules of m are essentially retractable nonsingular modules over r modulo their annihilators. (ii)  a non-zero module  is essentially retractable if and on...

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...

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 ...

We will use commutators to provide decompositions of 3×3 matrices as sums whose terms satisfy some polynomial identities, and we apply them bounded linear operators endomorphisms free modules infinite rank. In particular it is proved that every operator an infinite-dimensional complex Hilbert space a sum four automorphisms order 3 simple ring obtained quotient the endomorphism vector modulo its...

For any nite volume hyperbolic 3-manifold M we use ideal tri-angulation to deene an invariant (M) in the Bloch group B(C). It actually lies in the subgroup of B(C) determined by the invariant trace eld of M. The Chern-Simons invariant of M is determined modulo rationals by (M). This implies rationality and | assuming the Ramakrishnan conjecture | ir-rationality results for Chern Simons invariants.