唯一分解环的元素的模g同因分类

The Mod g Equi-divisor Classification between the Elements of a Factorial Ring

唯一分解环I的一个给定元g确定了I的元素间的一个等价关系~:a,b∈I,a^b当且仅当(a;g)=(b;g),从而得到I的一个分类I(g),a所在的类记为[a].得到了如下主要结果:当g≠0时,I(g)={[d1],[d2],…,[dT(g)]},其中T(g)是g的互不相伴的因子个数,d1,d2,…,dT(g)是g的T(g)个互不相伴的元;g∈I,有|I(g)|≤|I/(g)|;I(g)关于乘法[a][b]=[ab]作成以[1]为单位元的交换半群,且除[1]外其余的元都没有逆元.

A given element g of a factorial ring I determines an equivalence relation between the elements of I～：（V）a,b∈I,a～b ,if （ a ; g） = （ b ; g）. Hence, a classification of I referred to as I（g） can be obtained, and [ a ] denote the class which a belongs to. In this paper, the following results are gained： If g≠0, then/（g） = { [d1 ], [d2 ],… [dT（g）,]} , where T（g） is the number of non -associated factors of g, and d1 ,d2,…,dT（g） are non- associated factors of g; For V g ∈ I , ｜I（g）｜ ≤ ｜I/（g） ｜ exists; Based on the multiplication [ a ] [ b ] = [ab ] I （g） becomes a commutative semi - group, in which [ 1 ] is the unit element, and the other elements do not have inverse elements.