On the lifting problem for homogeneous ideals in polynomial rings

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...

On primitive ideals in polynomial rings over nil rings

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I [x] for some ideals I of R. All considered rings are associative but not necessarily have identities. Köthe’s conjecture states that a ring without nil ideals has no one-sided nil ideals. It is equivalent [4] to the assertion that polynomial rings over nil rings are Jaco...

On the Waring problem for polynomial rings.

In this note we discuss an analog of the classical Waring problem for C[x0,x1,...,x(n)]. Namely, we show that a general homogeneous polynomial p ∈ C[x0,x1,...,x(n)] of degree divisible by k≥2 can be represented as a sum of at most k(n) k-th powers of homogeneous polynomials in C[x0,x1,...,x(n)]. Noticeably, k(n) coincides with the number obtained by naive dimension count.

Lifting Problem in Codimension 2 and Initial Ideals

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...