The main purpose of this paper is to continue the study of uniform strong primeness on fuzzy setting. A pure fuzzy notion of this structure allows us to develop specific fuzzy results on USP (uniformly strongly prime) ideals over commutative and noncommutative rings. Besides, the differences between crisp and fuzzy setting are investigated. For instance, in crisp setting an ideal I of a ring R ...

This article concerns commutative factor rings for ideals contained in the center. A ring $R$ is called CIFC if $R/I$ some proper ideal $I$ of with $I\subseteq Z(R)$, where $Z(R)$ center $R$. We prove that (i) a $R$, $W(R)$ contains all nilpotent elements (hence Köthe's conjecture holds $R$) and $R/W(R)$ reduced ring; (ii) strongly bounded $R/N_*(R)$ $0\neq N_*(R)\subseteq (resp., $N_*(R)$) Wed...

Throughout this paper, R will represent an associative ring with center Z(R). A ring R is n-torsion free, where n > 1 is an integer, in case nx = 0, x ∈ R implies x = 0. As usual the commutator xy− yx will be denoted by [x, y]. We will use basic commutator identities [xy,z] = [x,z]y + x[y,z] and [x, yz] = [x, y]z+ y[x,z]. Recall that a ring R is prime if aRb = (0) implies that either a = 0 or b...

Several ideal-lattice-based cryptosystems have been broken by recent attacks that exploit special structures of the rings used in those cryptosystems. The same structures are also used in the leading proposals for post-quantum lattice-based cryptography, including the classic NTRU cryptosystem and typical Ring-LWE-based cryptosystems. This paper proposes NTRU Prime, which tweaks NTRU to use rin...

Much is known about the adele ring of an algebraic number field from perspective harmonic analysis and class theory. However, its ring-theoretical aspects are often ignored. Here, we present a description prime spectrum this study some topological properties these ideals. We also how they behave under separable extensions base give indication can be applied in rings not fields.

In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime i...

Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring an Abhyankar function field over perfect ground prime characteristic is Frobenius split. We show splitting sufficiently well-behaved center lifts to ring. also investigate properties valuations centered on arbitrary Noetherian domains characteristic. In contrast [arXiv:1507.06009], this paper emphasizes ...

For a module-finite algebra over commutative noetherian ring, we give complete description of flat cotorsion modules in terms prime ideals the algebra, as generalization Enochs' result for ring. As consequence, show that pointwise Matlis duality gives bijective correspondence between isoclasses indecomposable injective left and right modules. This is an explicit realization Herzog's homeomorphi...

‎let $r$ be a $*$-prime ring with center‎ ‎$z(r)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $r$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $r$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎suppose that $u$ is an ideal of $r$ such that $u^*=u$‎, ‎and $c_{sigma,tau}={cin‎ ‎r~|~csigma(x)=tau(x)c~mbox{for~all}~xin r}.$ in the present paper‎, ‎it is shown that...

the notions of quasi-prime submodules and developed  zariski topology was introduced by the present authors in cite{ah10}. in this paper we use these notions to define a scheme. for an $r$-module $m$, let $x:={qin qspec(m) mid (q:_r m)inspec(r)}$. it is proved that $(x, mathcal{o}_x)$ is a locally ringed space. we study the morphism of locally ringed spaces induced by $r$-homomorphism $mrightar...