The Prime Ideas Of PI-Algebra

Modern mathematics, one of the features of the study is hoped to find a large have a common characteristic algebraic structure, and the PI-algebra satisfies this condition, the vast majority of algebraic system is known as PI-algebra, in particular, all of the matrix algebra are PI-algebra. And research a algebraic system, we often want to start from simple question, this article will be a series of noncommutative algebra ideals problem into commutative algebra problem, through the ideas of center to depict the ideas of algebra itself, furthennore we discuss the prime ideas and max ideas between the algebra and it center.One of the central concerns to study PI-rings has been the behavior of the prime ideals. In this article we give a unified method for obtaining the major theorems on chains of prime ideals inaffine rings。Here is a sketch of our method:Let R be a prime PI-ring of pid(R)=n; then by Posner’s theorem R has a classicalring of quotients Q(R), which is a finite-dimensional central simple algebra.Therefore, the elements of R have characteristic polynomials. NOW let TR be the subring of Q(R) generated by R and all the coefficients of the characteristic polynomials of the elements of R. The study of TR and its close connection to R form the body of our work.First TR and R possess a common ideal V, and a prime ideal P in R contains I’if and only if pid(R/P)<pid(R). It also turns out there is a one-to-onecorrespondence between prime ideals in A not containing V and those of TR notcontaining I’. We proceed to study the general situation of two rings possessinga common ideal in Theorems7.5and7.6.Second, TR is integral over its center, and we investigate the behavior ofprime ideals between TR and its center (Theorems8.1and8.3). Hence, we study the prime ideals of R through those of TR which, in turn, are closely related to those of its center.For affine rings R。...