Factorization of prime ideal extensions in number rings

Factorization of Prime Ideal Extensions in Number Rings

Following an idea of Kronecker, we describe a method for factoring prime ideal extensions in number rings. The method needs factorization of polynomials in many variables over finite fields, but it works for any prime and any number field extension. Introduction Let F c K be number fields, let ¿fp c <?* be their corresponding number rings, i.e., the integral closures of Z in F and K, respective...

Completely Prime Ideal Rings and Their Extensions

Let R be a ring and let I 6= R be an ideal of R. Then I is said to be a completely prime ideal of R if R/I is a domain and is said to be completely semiprime if R/I is a reduced ring. In this paper, we introduce a new class of rings known as completely prime ideal rings. We say that a ring R is a completely prime ideal ring (CPI-ring) if every prime ideal of R is completely prime. We say that a...

Commutative Regular Rings without Prime Model Extensions

It is known that the theory K of commutative regular rings with identity has a model completion K . We show that there exists a countable model of K which has no prime extension to a model of K'. If K and K ate theories in a first order language L, then K is said to be a model completion of K if K extends K, every model of K can be embedded in a model of K , and for any model A of K and models ...

T-Rough (Prime, Primary) Ideal and T-Rough Fuzzy (Prime, Primary) Ideal on Commutative Rings

The purpose of this paper is to introduce and discuss the concept of T-rough (prime, primary) ideal and T-rough fuzzy (prime, primary) ideal in a commutative ring . Our main aim in this paper is, generalization of theorems which have been proved in [6, 7, 11]. At first, T-rough sets introduced by Davvaz in [6]. By using the paper, we define a concept of T-rough ideal , T-rough quotient ideal an...

Extensions of Nilpotent P.P. Rings