摘要
定义了PMM环.环R称为PMM环,若对任何Morita相似于R的环S,存在m,n∈N,使得Mm(S)同构于Mn(R).证明了如下结果:环R是PMM环当且仅当任给 R的投射生成元P,存在m,n∈N以及R上的Picard投射生成元Q,使得Pm 同构于Qn.具有VBN性质的PMM环是T2 环;具有IBN性质的PM环是T1 环.若交换环R是PMM环,则R是不可分解的且R的Picard群是幂可除的.特别地,Dedekind整环 R是 PMM环当且仅当 R的Picard群是幂可除的.
The PMM rings are defined and studied in this paper.A ring R is called a PMM ring if for any ring S which is Morita similar to R,M m(S) is isomorphic to M n(R) for some n,m∈N. The following results are proved in this paper.A ring R is a PMM ring if and only if whenever given a progenerator P over R, there exist m,n∈N and some Picard progenerator Q over R such that P m is isomorphic to Q n. PMM rings with VBN property are just T 2-rings;and with IBN property are T 1-rings. If R is a commutative PMM ring,then R is indecomposable and the Picard group of R is power divisible.In particular,a Dedekind domain R is a PMM ring if and only if the Picard group of R is power divisible.
