环的投射生成元和K0群

设Ｒ是含幺结合环，Ｐｇ（Ｒ）是Ｒ的所有投射生成元的同构类组成的半群，Ｇｒ（Ｐｇ（Ｒ））是Ｐｇ（Ｒ）的Ｇｒｏｔｈｅｎｄｉｅｃｋ群，在本文中我们证明了Ｋ０（Ｒ）＝Ｇｒ（Ｐｇ（Ｒ））。由此我们得到对任意ＶＢＮ环Ｒ，存在环Ｓ满足Ｓ＾２＝Ｓ并且具有Ａｕｔ－Ｐｉｃ性质，最后我们给出了环的一个分类，并且用Ｐｇ（Ｒ）的周期性对它作了描述。

PROGENERATORS AND K0-GROUPS OF RINGS

Let R be an associative ring with identity. Denote Pg〈R〉 for the semigroup of isomorphism classes of progenerators of R, and Gr(Pg〈R〉) for the Grothendieck group of Pg〈R〉.In this paper we prove that K0(R)≌Gr(Pg〈R〉). As an application, we obtain that for any VBN(i.e,non IBN) ring R, K0(R) is isomorphic to an ideal of Pg〈R〉. Then we prove that if R is a VBN ring,there exists a ring S such that S2≌S and S has the Aut-Pic property. Finally, we give a classification of rings and describe them by the periodicity of Pg〈R〉.