一元多项式环与中国剩余定理

Monadic Polynomial Ring and the Chinese Remainder Theorem

摘要: 本文对一般环R上的一元多项式环R[x]展开了研究.首先, 本文引入了R[x]中的一元多项式之间左(或右) 带余除法以及左(或右)辗转相除法,并给出了两个多项式能够进行左(或右)带余除法以及左(或右)辗转相除法的条件. 其次, 本文通过左(或右)辗转相除法引入了一元多项式有序对(f(x),g(x))伪互素这一概念,并证明了伪互素蕴含了理想的互素.再者, 利用伪互素的概念, 本文在非交换一元多项式环R[x]证明了一类左R[x]-模同态ϕ^Π的存在性.在本文的最后部分, 我们提供了一个关于ϕ^Π的理论应用,并指出ϕ^Π在R为交换幺环的情况下就是R[x]上的中国剩余定理.

Abstract: This paper conducts a study on the monadic polynomial ring R[x] over a ring R. First of all, we introduce the left (or right) division with remainder and the left (or right) Euclidean algorithm between two univariate polynomials in R[x], and provide a condi-tion under which two polynomials can perform left (or right) division with remainder and left (or right) Euclidean algorithm. Secondly, by utilizing the left (or right) Euclidean algorithm, the paper introduces the concept of pseudo-coprimality for ordered pairs of univariate polynomials (f(x), g(x)), and proves that pseudo-coprimality implies the coprimality of ideals. Furthermore, by using pseudo-coprime, the paper demon-strates the existence of a left R[x]-module homomorphism ϕ^Π in the non-commutative univariate polynomial ring R[x]. In the final part of the paper, we provide a theoretical application ϕ^Π and point out that it corresponds to the Chinese Remainder Theorem on R[x] when R is a commutative ring with unity.