关于ω阶Euclid模(英文)

ω-Stage Euclidean Modules

左R-模M称为ω阶Euclid模,如果对任何a,b∈M,a≠0,存在k阶可除链(k∈N),使得ф(mk)<ф(a),其中ф:M→N∪{0}且满足:ф(m)≥0对任何m∈M;ф(m)=0当且仅当m=0.文章证明了:ω阶Euclid模存在着有限的可除链;每一个单模是ω阶Euclid模且ω阶Euclid模的子模是循环的;ω阶Euclid模的同态核与同态象仍是ω阶Euclid模,但后者的逆命题不成立,并构造了一个适当的反例.

A left R-module M is called co-stage Euclidean module, provided that if any arbitrary elements a,b∈M,a≠0, there exists a k-stage division chain for some k, such that Ф（mk）〈Ф（a） when Ф：M→NU{0} with Ф（m）≥0 for all m∈M and Ф（m）=0 if and only if m=0. The article proves that every ω-stage Euclidean module has a terminating k-stage division chain. Every simple module is ω-stage Euclidean and every submodule of an of-stage Euclidean module is cyclic. Homomorphic kernels and homomorphic images of an ω-stage Euclidean module are ω-stage Euclidean, but the inverse proposition of the latter one is false. The paper constructed a suitable counterexample as well.