Mathematics

We provide an expression for the conductor c(Δ) of a semimodule Δ of a numerical semigroup Γ with two generators in terms of the syzygy module of Δ and the generators of the semigroup. In particular, we deduce that the difference between the conductor of the semimodule and the conductor of the semigroup is an element of Γ , as well as a formula for c(Δ) in terms of the dual semimodule of Δ .

Recently, nearly complete intersection ideals were defined by Boocher and Seiner to establish lower bounds on Betti numbers for monomial ideals (arXiv:1706.09866). Stone and Miller then characterized nearly complete intersections using the theory of edge ideals (arXiv:2101.07901). We extend their work to fully characterize nearly complete intersections of arbitrary generating degrees and use this characterization to compute minimal free resolutions of nearly complete intersections from their degree 2 part.

Using an old example of Nagata, we construct a Noetherian ring of prime characteristic p, whose Frobenius morphism is locally finite, but not finite.

The principal minors of the Toeplitz matrix ( x i−j+1 ) 1≤i,j,≤n , where x 0 =1, x k =0 if k≤−1 , directly determine an involution of the polynomial ring R[ x 1 ,..., x n ] over any commutative ring R .

We investigate the behavior of Cohen-Macaulay defect undertaking tensor product with a perfect module. Consequently, we study the perfect defect of a module. As an application, we connect to associated prime ideals of tensor products.

Let u be in p∈Assh(R) . We present several situations for which (0:u) is (not) in an ideal generated by a system of parameters. An application is given.

Let k be a field. In this paper we will show that any factorial A 1 -form A over any k -algebra R is trivial, if A has a retraction to R .

We study the ring extensions R \subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples are also discussed to strengthen the results.

We deal with the complete-intersection property of maximally differential ideals. Also, we connect the Gorenstein homology of derivations to the Gorenstein property of the base rings. These equipped with some applications.

Let R be a regular local ring of dimension d and H d−1 J (R) be the (d−1) -th local cohomology module supported at the ideal J . The second vanishing theorem for the local cohomology or the SVT is well known in regular local rings containing a field and for complete unramified regular local ring of mixed characteristic. It states that under certain conditions on J (see below for detail), H d−1 J (R) vanishes if and only if the punctured spectrum R/J is connected. In this paper, we extend the result of SVT to complete ramified regular local ring only for the extended ideals. When punctured spectrum is not connected, we find that the punctured spectrum of the corresponding unramified regular local ring also is also disconnected and having same number of connected components. Moreover, we show that the Matlis duals of those local cohomology modules support the Conjecture 1 and the result of Corollary 1.2 of \cite{L-Y}.