In this paper we investigate the sufficiency criteria which guarantee the classical localization of a bounded ring at its prime ideals.

Local distinguishability of sets generalized Bell states (GBSs) is investigated. We first clarify the conditions such that a set GBSs can be locally transformed to certain type GBS easily distinguishable within local operations and one-way classical communication. then show that, if space dimension $d$ prime, these are necessary sufficient for in $\mathbb{C}^d \otimes \mathbb{C}^d$ distinguisha...

Let \(c \geq 2\) be any fixed real number. Matomäki [4] inverstigated the set of \(A > 1\) such that integer part \( A^{c^k}\) is a prime number for every \(k \in \mathbb{N}\). She proved uncountable, nowhere dense, and has Lebesgue measure 0. In this article, we show Hausdorff dimension 1.

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1...

We study the “q-commutative” power series ring R := kq[[x1, . . . , xn]], defined by the relations xixj = qijxjxi, for mulitiplicatively antisymmetric scalars qij in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In par...

We present two methods for the construction of quantum circuits for quantum errorcorrecting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n− k)) gates. The running time of the classical algorithm to compute the quantum circuit is ...

We give a simple formula for the essential dimension of a finite pseudoreflection group at a prime p and determine the absolute essential dimension for most irreducible pseudo-reflection groups. We also study the “poor man’s essential dimension” of an arbitrary finite group, an intermediate notion between the absolute essential dimension and the essential dimension at a prime p.