Semiclean环及其扩张

         

摘要

Clean rings are generalized and semiclean rings are investigated. Some properties of semiclean rings are discussed. It shows that the followings are equivalent : (1)R is a semiclean ring; (2)Its formal power series ring R[[x]] is a semiclean ring ; (3)Its skew power series ring R[[x;a]] is a semiclean ring. Finally, the following result is proved: Let R be a ring and (S,≤) is a strictly ordered monoid satisfying the condition that 0≤s for every s €: S. Then [[Rs,≤]]is a semiclean ring if and only if R is a semiclean ring.%对clean环进行了推广,研究了semiclean环,讨论了semiclean环的几个重要性质;证明了(1)R是semiclean环;(2)R上的形式幂级数环R[[x]]是semiclean环;(3)R上的斜幂级数环R[[x;α]]是semiclean环等价;最后证明了,如果R是环,(S,≤)是严格偏序幺半群且对任意s∈S,都有0≤s,则[[RS,≤]]是semiclean环当且仅当R是semiclean环.

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