In this paper we prove analogs for the case of bigraded polynomial rings of theorems about regularity and saturation of ideals in simply graded polynomial rings.

This note announces a number of results on the structure of differential modules over differential rings, where differential ring means a ring with a family of derivations and differential module means a module having a family of operators compatible with the derivations of the ring. To fix notation, throughout the paper we let A denote an associative ring, M = AM an 4-module, k the correspondi...

The Ring-LWE problem, introduced by Lyubashevsky, Peikert, and Regev (Eurocrypt 2010), has been steadily finding many uses in numerous cryptographic applications. Still, the Ring-LWE problem defined in [LPR10] involves the fractional ideal R∨, the dual of the ring R, which is the source of many theoretical and implementation technicalities. Until now, getting rid of R∨, required some relatively...

In this study we explore the subrings in trigonometric polynomial rings. Consider the rings T and T ′ of real and complex trigonometric polynomials over the fields R and its algebraic extension C respectively ( see [6]). We construct the subrings T0 of T and T ′ 0, T ′ 1 of T ′. Then T0 is a BFD whereas T ′ 0 and T ′ 1 are Euclidean domains. We also discuss among these rings the Condition : Let...

We prove that Zpn and Zp[t]/(t) are polynomially equivalent if and only if n ≤ 2 or p = 8. For the proof, employing Bernoulli numbers, we provide the polynomials which compute the carry-on part for the addition and multiplication in base p. As a corollary, we characterize finite rings of p elements up to polynomial equivalence.

The main object of study in this paper is the completion Z[q] = lim ←−n Z[q]/((1−q)(1−q) · · · (1−q)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3spheres with values in Z[q] announced by the author, which unifies all the sl2 Witten-Reshetikhin-Turaev invariants at various roots of unity. We show that any element of Z[q] is uniquely determine...

In this paper we investigate how algorithms for computing heights, radicals, unmixed and primary decompositions of ideals can be lifted from a Noetherian commutative ring R to polynomial rings over R. It is a standard problem in mathematics to study which properties of a mathematical structure are preserved in derived structures. A typical result of this kind is the Hilbert Basis Theorem which ...

We prove the following form of Dirichlet’s theorem for polynomial rings in one indeterminate over a pseudo algebraically closed field F . For all relatively prime polynomials a(X), b(X) ∈ F [X] and for every sufficiently large integer n there exist infinitely many polynomials c(X) ∈ F [X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.

Let $R$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $R$ and $f(X)=a_0+a_1X+cdots+a_nX^n$ be a nonzero skew polynomial in $R[X;alpha]$. It is proved that if there exists a nonzero skew polynomial $g(X)=b_0+b_1X+cdots+b_mX^m$ in $R[X;alpha]$ such that $g(X)f(X)=c$ is a constant in $R$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $R$ such tha...

In analogy to cyclic codes, we study linear codes over finite fields obtained from left ideals in a quotient ring of a (non commutative) skew polynomial ring. The paper shows how existence and properties of such codes are linked to arithmetic properties of skew polynomials. This class of codes is a generalization of the θ-cyclic codes discussed in [1]. However θ-cyclic codes are performant repr...