Coherence of polynomial rings and bounds in polynomial ideals

On annihilator ideals in skew polynomial rings

This article examines annihilators in the skew polynomial ring $R[x;alpha,delta]$. A ring is strongly right $AB$ if everynon-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property ($A$) and the conditions asked by P.P. Nielsen. We assume that $R$ is an ($alpha$,$delta$)-compatible ring, and prove that, if $R$ is ni...

A Property of Ideals in Polynomial Rings

Every ideal in the polynomial ring in n variables over an infinite field has a reduction generated by n elements. Eisenbud and Evans [2] proved that every ideal in k[Xx,...,Xn] can be generated up to radical by n elements (where k is a field). Avinash Sathaye [7] and Mohan Kumar [5] proved a locally complete intersection in k[ Xv ..., Xn] can be generated by n elements. In this short note we sh...

Associated Prime Ideals of Skew Polynomial Rings

In this paper, it has been proved that for a Noetherian ring R and an automorphism σ of R, an associated prime ideal of R[x, σ] or R[x, x−1, σ] is the extension of its contraction to R and this contraction is the intersection of the orbit under σ of some associated prime ideal of R. The same statement is true for minimal prime ideals also. It has also been proved that for a Noetherian Q-algebra...

Bounds and Definability in Polynomial Rings

We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor. In particular, we show that the module of syzygies of polynomials f1, . . . , fn ∈ R[X1, . . . , XN ] with coefficients in a Prüfer domain R can be generated by elements whose degrees are bounded by a number only depending on N ...

Differential Polynomial Rings of Triangular Matrix Rings