Existence Of Coefficient Subring for Transcendental Extension Ring

Hanan Abdulaziz Alolaiyan

doi:10.24297/jam.v13i3.6088

Abstract

As a consequence of Cohen's structure Theorem for complete local rings that every _nite commutative ring R of characteristic pn contains a unique special primary subring R0 satisfying R/J(R) = R0/pR0: Cohen called R0 the coe_cient subring of R. In this paper we will study the case when the ring is a transcendental extension local artinian duo ring R; we proved that even in this case R will has a commutative coe_cient subring.

Highlights

Introduced the concept of an inertial sub algebra of algebra over a Hensel ring
Singh have generalized the result to the existence of coefficient subring of a locally finite R algebraA, where R is a commutative chain ring such that J (R) = R J ( A), A/J ( A) countably generated separable algebraic field extension
The existence and structure of R0 for finite commutative local ring R was known to Krull as early as 1924 [11], p

Summary

Introduction

Introduced the concept of an inertial sub algebra of algebra over a Hensel ring. The concept of an inertial sub algebra is analogous to that of a coefficient subring Y. Singh have generalized the result to the existence of coefficient subring of a locally finite R algebraA , where R is a commutative chain ring such that J (R) = R J ( A), A/J ( A) countably generated separable algebraic field extension. Corbas [8] manages to characterize coefficient subring of a finite ring as a direct sum of full matrix rings over Galois rings.Y. Alkhamees and S. Singh [1] have generalized the results on the existence of coefficient subrings of finite local rings to locally finite R algebra A , where R is commutative chain ring such that J (R) = R J ( A), A/J ( A) countably generated separable algebraic field extension. Ring of all algebraic elements in R over the prime subring

Preliminaries

Coefficient subrings