环的投射生成元和K_0群(英文)

PROGENERATORS AND K_0-GROUPS OF RINGS

设Ｒ是含么结合环，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Ｋ_０（Ｒ）＝ Ｇｒ（Ｐｇ＜Ｒ＞）．由此我们得到对任意ＶＢＮ环（即非ＩＢＮ环）Ｒ，存在环Ｓ满足Ｓ~２＝Ｓ并且Ｓ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．

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 K, (R) = Gr(Pg<R> ). As an application, we obtain that for any VBN (i. e, non IBN) ring R, K, (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 S^2 = S and S has the Ant-Pic property. Finally, we give a classification of rings and describe them by the periodicity of Pg(R}.