We introduce graded $${\mathbb {E}}_{\infty }$$ -rings and modules over them, study their properties. construct projective schemes associated to connective {N}}$$ -graded in spectral algebraic geometry. Under some finiteness conditions, we show that the $$\infty $$ -category of almost perfect quasi-coherent sheaves a scheme $$\text { {Proj}}\,(A)$$ -ring A can be described terms $${{\mathbb {Z}...

In this paper, we study the linear codes over the commutative ring R = Z4 + vZ4, where v2 = v. We define the Gray weight of the elements of R and give a Gray map from Rn to Z2n 4 , which lead to the MacWillams identity of the linear code over R. Some useful results on self-dual code over R are given. Furthermore, the relationship between some complex unimodular lattices and Hermitian self-dual ...

Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors. Fix a prime p ∈ Z and f ∈ (Z/pZ)[x] any nonzero polynomial of degree d whose coefficients are not all divisible by p. For the case n = 2, we prove a new effi...

In this paper, we study a relative two-weight Z2Z4-additive codes. It is shown that the Gray image of a two-distance Z2Z4-additive code is a binary two-distance code and that the Gray image of a relative two-weight Z2Z4-additive code, with nontrivial binary part, is a linear binary relative two-weight code. The structure of relative two-weight Z2Z4-additive codes are described. Finally, we disc...

In this paper, we study one-Lee weight and two-Lee weight codes over Z2Z2[u], where u = 0. Some properties of one-Lee weight Z2Z2[u]-additive codes are given, and a complete classification of one-Lee weight Z2Z2[u]-additive formally self-dual codes is obtained. The structure of two-Lee weight projective Z2Z2[u] codes are determined. Some optimal binary linear codes are obtained directly from on...

In this paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of coordinate ring a complement hyperplane arrangement. We give presentation these by generators and defining relations. This was used Holler I. Kriz calculate $\mathbb{Z}$-graded coefficients localizations ordinary $RO((\mathbb{Z}/p)^n)$-graded equivariant cohomology at giv...

This note is devoted to the study of totally archimedean rings of regular functions. We easily extend Schm udgen theorem to this class of rings. Moreover, we show that, in such rings, every totally positive element is a sum of even powers of totally positive elements, and hence is a sum of even powers of units.

Let $R$ be a commutative ring and $mathbb{A}(R)$ be the set of all ideals with non-zero annihilators. Assume that $mathbb{A}^*(R)=mathbb{A}(R)diagdown {0}$ and $mathbb{F}(R)$ denote the set of all finitely generated ideals of $R$. In this paper, we introduce and investigate the {it finitely generated subgraph} of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_F(R)$. It is the (undi...

Abstract For every $m\in \mathbb {N}$ , we establish the equidistribution of sequence averaged pullbacks a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under $m$ th order derivatives iterates polynomials $f\in {C}[z]$ degree $d>1$ towards harmonic filled-in Julia set f with pole $\infty $ . We also non-archimedean and arithmetic counterparts using potential theory on Berk...