半正则理想与Q_(0)-Noether环

Semiregular Ideals and Q_(0)-Noetherian Rings

设R是交换环,引入Lucas模与模的Lucas包络的概念,证明理想I的Lucas包络Q_(0)(I)≠Q_(0)(R)当且仅当I是R的非半正则理想;也引入Q_(0)-Noether环的概念,并证明R是Q_(0)-Noether环当且仅当多项式环R[x_(1),x_(2),…,x_(n)]是Q_(0)-Noether环.

Let R be a commutative ring.In this paper,Lucas modules and Lucas envelopes of modules are introduced.It is proved that the Lucas envelope Q_(0)(I)of an ideal I of R is not equal to Q_(0)(R)if and only if I is non-semiregular ideal of R.As an application,we also introduce the concept of Q_(0)-Noetherian ringss and provide the relative Hilbert basis theorem of Q_(0)-Noetherian rings,i.e.,R is a Q_(0)-Noetherian ring if and only if the polynomial ring R[x_(1),x_(2),…,x_(n)]is a Q_(0)-Noetherian ring.