主理想整环的几个等价刻画

SOME EQUIVALENT CONDITIONS OF THE PRINCIPAL IDEAL DOMAINS

若R为唯一分解整环 ,则下列两条件之一皆等价于R为主理想整环 :1)任给A∈Mm ,n(R) ,则必有R上的可逆矩阵P ,Q使PAQ =diag{d1,d2 ,… ,di,0 ,… ,0 } ; 2 )M是有限生成R_模 ,则存在唯一的t∈Z ,r∈R ,使M R(t) R/ (r1) … R/ (rs) ri|ri+1,i=1,… ,s- 1。

In this paper,based on the theory of the structure theorem for finitely generated modules over p.i.d.,the auther shows several equivalent conditions of the principal ideal domains.