幺半群的内射类及投射类刻画

摘要

In this paper we initiate the study of injectivity classes of S-acts over monoids, analogous to the ring theoretical notion of injectivity classes of general modules. We also introduce the dual notion of projectivity classes of S-acts and apply these notions to characterize certain classes of monoids. Among other results, we show that a monoid is completely injec-tive if and only if it is completely quasi-injective. Furthermore, if I is a projectivity class of S'-acts over a perfect monoid S, then S is completely projective if and only if S is completely quasi-projective if and only if 7 is the category of right S-acts over S.%类比于一般环上模的内射类,定义了幺半群上的S-系的内射类和投射类,并利用它们刻画了几类特殊的幺半群.证明了完全内射幺半群和完全拟内射幺半群是等价的.并且证明了对于标致幺半群S,它是完全投射的当且仅当它是完全拟投射的当且仅当它上面的投射S-系构成了一个投射类.