所有循环平坦系满足条件(P)的PSF幺半群

PSF MONOIDS OVER WHICH ALL CYCLIC FLAT RIGHT ACTS SATISFY CONDITION (P)

本文讨论所有循环平坦系满足条件（Ｐ）的幺半群的“元素——理想”特征问题，该问题至今仍未获解决． 在Ｓ是左ＰＳＦ幺半群的条件下，本文证明了所有循环平坦右Ｓ－系满足条件（Ｐ）当且仅当Ｓ的任意元ｘ 或者是右可消元，或者是右零元，也当且仅当对Ｓ的任意真右理想Ｉ，或存在ａ∈Ｉ－ Ｉａ，或Ｉ中的所有元素均为右零元，该结果改进并推广了［４］、［７］、［８］。

This paper devoted to consider monoids over which all cyclic flat right acts satisfy condition (P).If S is a left PSF monoid,then it is proved that all cyclic flat right S acts satisfy condition (P) if and only if every element of S is either right zero or right cancellative if and only if for every proper right ideal I of S,either there exists a∈I-Ia or every elementof I is right zero.As corollaries,some results of [4],[7],[8]and [15] are generalized. This paper devoted to consider monoids over which all cyclic flat right acts satisfy condition (P).If S is a left PSF monoid,then it is proved that all cyclic flat right S acts satisfy condition (P) if and only if every element of S is either right zero or right cancellative if and only if for every proper right ideal I of S,either there exists a∈I-Ia or every elementof I is right zero.As corollaries,some results of [4],[7],[8]and [15] are generalized.