Journal of Pure and Applied Algebra ( IF 0.7 ) Pub Date : 2021-03-16 , DOI: 10.1016/j.jpaa.2021.106744 L. Klingler , K.A. Loper , W.Wm. McGovern , M. Toeniskoetter
Recall that a ring is said to be a clean ring if every element can be expressed as the sum of a unit and an idempotent. In one variant of this definition, a ring is said to be a semi-clean ring if every element can be expressed as the sum of a unit and a periodic element. Ye's Theorem [12] states that the group ring is semi-clean, where p is a prime integer and is a cyclic group of order 3. In this article, we generalize Ye's Theorem by demonstrating that, if R is a local ring, then the group ring is semi-clean if and only if G is a torsion abelian group.
中文翻译:
半清洁组环
回想一下,如果每个元素都可以表示为一个单位和一个幂等数之和,则将一个环称为干净环。在该定义的一个变体中,如果每个元素都可以表示为一个单元和一个周期元素之和,则该环被称为半清洁环。叶定理[12]指出群环是半干净的,其中p是一个质数整数,是阶数为3的一个循环群。在本文中,我们通过证明,如果R是一个局部环,则该群环是对叶氏定理的推广。当且仅当G是一个扭转阿贝尔群时,它是半清洁的。