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中文摘要：
介绍了强π-正则一般环（未必有单位元）的概念并考虑了它的一些扩张．给出了强π-正则一般环的2个等价刻画，即，是强π-正则一般环当且仅当对于每个x∈I，存在n≥1以及y，z∈I，使得x^n=x^n＋1y=zx^n＋1当且仅当I中的每个元都是强π-正则的．还考虑了强π-正则一般环上的上三角矩阵一般环和平凡扩张，证明了强π-正则一般环上的上三角矩阵一般环仍是强π-正则的并且其平凡扩张是强clean的．
英文摘要：
The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.
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Larson-Sweedler theorem and some properties of discrete type in ($G$-cograded) multiplier Hopf algeb
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