For every ring R, we construct a ringed space NCSpec(R) and for every ring homomorphism R → S, a morphism of ringed spaces NCSpec(S) → NCSpec(R). We show that this gives a fully faithful contravariant functor from the category of rings to a category of ringed spaces. If R is a commutative ring, we show that NCSpec(R) can be viewed as a natural completion of Spec(R). We then explain how the spac...

By the Quillen-Suslin theorem [Qui76, Sus76], we know that projective modules over a polynomial ring over a field are free. One way of saying this is, that if two projective modules of the same rank are stably isomorphic, then they are isomorphic. That, projective modules of given rank over polynomial rings are stably isomorphic was well known at the time Quillen and Suslin proved their theorem...

We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error–locator polynomial is the unique monic minimal polynomial (shortest linear recurrence) of the syndrome sequence and that it can be obtained by Algorithm MR of Norton. When R is a local ring, we show that the syndrome sequence may have ...

In this paper we show that a commutative semisimple ring is always a Smarandache ring. We will also give a necessary and sufficient condition for group algebra to be a Smarandache ring. Examples are provided for justification.

We give a dimension bound on the irreducible components of the characteristic variety of a system of linear partial di.erential equations de0ned from a suitable 0ltration of the Weyl algebra An. This generalizes an important consequence of the fact that a characteristic variety de0ned from the order 0ltration is involutive. More explicitly, we consider a 0ltration of An induced by any vector (u...

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y], which establishes a conjecture of Haiman and Sturmfels.

Let K[x] be a polynomial algebra in a variable x over a commutative Q-algebra K, and Γ′ be the monoid of K-algebra monomorphisms of K[x] of the type σ : x 7→ x + λ2x 2 + · · · + λnx , λi ∈ K, λn is a unit of K. It is proved that for each σ ∈ Γ ′ there are only finitely many distinct decompositions σ = σ1 · · · σs in Γ ′. Moreover, each such a decomposition is uniquely determined by the degrees ...

A commutative associative algebra A over C with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for A as a vertex algebra and the modules for A as an associative algebra are not well understood. In this paper, I give the classification of finite-dimensional indecomposable untwisted or twisted modules for the polynomial ring in one v...

In this article, we carry out the investigation for regular sequences of symmetric polynomials in the polynomial ring in three and four variable. Any two power sum element in C[x1,x2, . . . ,xn] for n ≥ 3 always form a regular sequence and we state the conjecture when pa, pb, pc for given positive integers a < b < c forms a regular sequence in C[x1,x2,x3,x4]. We also provide evidence for this c...

In 1971, Ruzsa conjectured that if f : N ? Z with ( n + k ) ? mod for every , ? and = O ? < e then is a polynomial. this paper, we investigate the analogous problem ring of polynomials over finite field using Polynomial Method in combinatorics.