摘要
理想A称为ω阶Euclid理想,如果对任何a,b∈A,a≠0,有k阶可除链(k∈N),使得φ(rk)<φ(a),其中φ:A→N∪{0}且满足:φ(x)≥0对任何x∈A;(x)=0当且仅当x=0.文章建立了ω阶Euclid理想与有限可除链之间的充分必要关系,证明了ω阶Euclid理想中两个元素(至少有一个不为零)存在最大公因子和每一个ω阶Euclid理想是主理想,构造了一个适当的例子,证明了ω阶Euclid理想上每一个n阶矩阵能通过初等变换简化为标准对角阵.
An ideal A of a ring is called ω-stage Euclidean ideal provided that if any arbitrary elements a,b∈A,a≠0,there exists a k-stage division chain for some k such that （rk）（a）,where φ：A→N∪{0} with φ（x）≥0 for all x∈A and φ（x）=0 if and only if x=0.The article has established necessary and sufficient relations between ω-stage Euclidean ideals of a ring and terminating division chains.It is shown that two elements（one of them is a nonzero element at least） of an ω-stage Euclidean ideal have the greatest common divisor and every ω-stage Euclidean ideal of a ring is principal.A suitable example is constructed as well.Every n×n matrix over ω-stage Euclidean ideals can be reduced to a canonical diagonal matrix by elementary transformation.
出处
《杭州师范大学学报（自然科学版）》
CAS
2010年第4期281-285,共5页
Journal of Hangzhou Normal University(Natural Science Edition)
