某些多项式环的自同构群与可逆理想  

Automorphism Groups and Invertible Ideals of Certain Polynomial Domains

作　　者：陈志力 胡葵 CHEN Zhili;HU Kui(College of Science,Southwest University of Science and Technology,Mianyang 621010,Sichuan)

机构地区：[1]西南科技大学数理学院,四川绵阳621010

出　　处：《四川师范大学学报(自然科学版)》2025年第3期417-421,共5页Journal of Sichuan Normal University(Natural Science)

基　　金：国家自然科学基金(12101515)。

摘　　要：域上的多项式环是非常重要且应用广泛的整环,研究某类比较特殊的多项式环上的一些性质,也是学者们一直在做的工作.设F_(q)是q个元素的有限域,对多项式环F_(q)+x^(2)F_(q)[x]的自同构群与可逆理想这2个方面分别进行研究.对于多项式环F_(q)+x^(2)F_(q)[x]的自同构群,证明Aut(F_(q)+x^(2)F_(q)[x])是阶为q-1的循环群.对于多项式环F_(q)+x^(2)F_(q)[x]的可逆理想,确定Pic(F_(q)+x^(2)F_(q)[x])的阶数是q,并确定了环F_(q)+x^(2)F_(q)[x]的理想一定是2-生成的.在此基础上,对F_(q)+x^(2)F_(q)[x]的理想进行分类讨论,得出F_(q)+x^(2)F_(q)[x]的可逆理想一定同构于(1+ax,x^(2)),a∈F_(q).Polynomial domains over fields are very important and widely used domains, and studying some properties on some particular class of polynomial domains is also the work that many scholars have always been doing. Let F_(q) be the finite field of q elements. In this paper, the automorphism group and invertible ideals of the polynomial domain F_(q)+x^(2)F_(q)[x] are studied. For the automorphism group of the polynomial domain F_(q)+x^(2)F_(q)[x] , this paper proves that Aut (F_(q)+x^(2)F_(q)[x]) is a cyclic group of order q-1 . For invertible ideals of the polynomial domain F_(q)+x^(2)F_(q)[x] , it is proved that the order of Pic (F_(q)+x^(2)F_(q)[x]) is q , and it is also determined that every ideal of the domain F_(q)+x^(2)F_(q)[x] must be 2-generated. On this basis, ideals of the domain F_(q)+x^(2)F_(q)[x] are classified, and it is concluded that any invertible ideal of F_(q)+x^(2)F_(q)[x] must be isomorphic to (1+ax,x^(2)) , a∈F_(q) .

关 键 词：自同构群 整环上的皮卡群 可逆理想 多项式环 

分 类 号：O153.3[理学—数学]