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 ...

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...

This article generalizes the well-known notion of generic forms to the algebra R 0 , introduced in 27]. For the total degree, then reverse lexicographic order, we prove that the initial ideal of an ideal generated by nitely many generic forms (in countably innnitely many variables) is nitely generated. This contrasts to the lexicographic order, for which initial ideals of generic ideals in gene...

Introduction Group algebras K[G] of poly cyclic-by-finite groups are easily-defined, interesting examples of right and left Noetherian rings. Since prime rings and prime ideals are the basic building blocks in the Goldie theory of Noetherian rings, the determination of the structure of the prime ideals of K[G] is certainly of importance. In a recent fundamental paper [8], Roseblade proved that ...

Mayr and Meyer [MM] found ideals with the doubly exponential ideal membership property. Further investigations of the doubly exponential properties of these ideals can be found in Bayer and Stillman [BS], Demazure [D], and Koh [K]. Following a question of Bayer, Huneke and Stillman, the author has investigated in [S1, S2, S3] the properties of the primary decompositions and the associated prime...