We show how tropical varieties of ideals I E K[x] over a field K with non-trivial valuation can always be traced back to tropical varieties of ideals π−1I E RJtK[x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that π−1I is generated by elements in R[t, x]. While doing so, we ...

We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$Cs). Insight from ring theory used obtain framework for prime, semiprime, and completely prime (thick) ideals an M$\Delta$C, ${\bf K}$, then associate K}$ topological space--the Balmer spectrum ${\rm Spc}{\bf K}$. (noncommutative) support ...

This article introduces quasi-ideals in semirings on weak nearness approximation spaces. Concepts and definitions are given to clarify the subject of quasi ideals Some basic properties also given. Furthermore, it is that definition upper-near ideals. And, examined relationship between upper near Therefore, features described this study will contribute greatly theoretical development theory.

Let I be an m-primary ideal in a Gorenstein local ring (A,m) with dimA = d, and assume that I contains a parameter ideal Q in A as a reduction. We say that I is a good ideal in A if G = ∑ n≥0 I n/In+1 is a Gorenstein ring with a(G) = 1−d. The associated graded ring G of I is a Gorenstein ring with a(G) = −d if and only if I = Q. Hence good ideals in our sense are good ones next to the parameter...

Let K be a number field and r an integer. Given an elliptic curve E , defined over K , we consider the problem of counting the number of degree two prime ideals of K with trace of Frobenius equal to r . Under certain restrictions on K , we show that “on average” the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard...

The foldness of PI( , ⊆, ⊆)BCK -ideals and PI( , , )BCK -ideals is considered. The fuzzy version of such notions is also discussed.

This report gives an overview of the main ideas in Chapter 9 of the studied book, about the dimension of a variety. After recalling some definitions and basic properties about projective varieties, homogeneous ideals and Gröbner bases, we will begin our study of the dimension of a variety with the observation of a geometric way to define the dimension of a variety defined by a monomial ideal. T...

Let I be a monomial squarefree ideal of a polynomial ring S over a field K such that the sum of every three different ideals of its minimal prime ideals is the maximal ideal of S, or more generally a constant ideal. We associate to I a graph on [s], s = |MinS/I|, on which we may read the depth of I. In particular, depthS I does not depend on char K. Also we show that I satisfies Stanley’s Conje...

In [23], Zadeh introduced the notion of fuzzy sets and fuzzy set operations. Since then, fuzzy set theory developed by Zadeh and others has evoked great interest among researchers working in different branches of mathematics. Semirings play an important role in studying matrices and determinants. Many aspects of the theory of matrices and determinants over semirings have been studied by Beasley...