Let K be a number field and OK be the ring of algebraic integers. We discuss the unique factorization of elements of OK into irreducibles and its use in solving Diophantine equations. We then proceed to prove the existence of the unique factorization of ideals of OK into prime ideals.

In this paper, we introduce the notion θ -Euclidean k–fuzzy ideal in semirings and to study the properties of the image and pre image of a θ -Euclidean k–fuzzy ideal in a semirings under epimorphism. Keywords—semiring, fuzzy ideal, k–fuzzy ideal,θ -Euclidean Lfuzzy ideal, θ -Euclidean fuzzy k–ideal, θ -Euclidean k-fuzzy ideal. 2000 AMS Classification: 16Y60, 13E05, 03G25.

generalizing the concepts of -fuzzy (left, right, lateral) ideals, -fuzzy quasi-ideals and   -fuzzy bi (generalized bi-) ideals in ternary semigroups, the notions of -fuzzy (left, right, lateral) ideals, -fuzzy quasi-ideals and -fuzzy bi (generalized bi-) in ternary semigroups are introduced and several related properties are investigated. some new results are obtained.

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with K. Petersen [8]. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices.

Let K be a field and X an m×n matrix of indeterminates over K. Let K[X] denote the polynomial ring generated by all the indeterminates Xij . For a given positive integer r ≤ min{m, n}, we consider the determinantal ideal Ir+1 = Ir+1(X) generated by all r + 1 minors of X if r < min{m, n} and Ir+1 = (0) otherwise. Let Rr+1 = Rr+1(X) be the determinantal ring K[X]/Ir+1. Determinantal ideals and ri...

Algebraic structures especially an ordered semigroups play a prominent role in mathematics with wide ranging applications in many disciplines such as control engineering, computer arithmetics, coding theory, sequential machines and formal languages. A theory of fuzzy sets in terms of fuzzy points on ordered semigroups can be developed. In this paper, we generalize the concept of (α, β)-fuzzy le...

INTRODUCTION Recently, the use of Groebner bases and Buchberger algorithm [BUC1,2,4] has been generalised from the case of commutative polynomials to finitely generated algebras R over a field k, R = k, s.t. for each i < j, for some cij ∈ k, for some commutative polynomial pij ∈ k[X1,...,Xn], one has xj xi cij xi xj = pij(x1,...,xn). The first results in this direction were due to Ga...

For a countably generated group K and a function on K taking values that are either nonnega-tive integers or 1, we determine in Theorem 8 equivalent conditions for there to exist a Dedekind domain D with class group K such that, for each k in K, exactly (k) maximal ideals of D belong to k.

Let J ⊂ S = K[x0, . . . , xn] be a monomial strongly stable ideal. The collection Mf(J) of the homogeneous polynomial ideals I, such that the monomials outside J form a K-vector basis of S/I, is called a J-marked family. It can be endowed with a structure of affine scheme, called a J-marked scheme. For special ideals J , J-marked schemes provide an open cover of the Hilbert scheme Hilbnp(t), wh...

A (hyper)graph G is called k-critical if it has chromatic number k, but every proper sub(hyper)graph of it is (k− 1)-colourable. We prove that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k−o(k))n edges. Furthermore, we show that every (k+1)-critical hypergraph on n vertices and without graph edges has at least (k−3/ 3 √ k)n edges. Both bounds diffe...