The Study On Non-semiregular Ideals And Q₀-Noetherian Rings

Let R be a commutative ring.Firstly,Lucas modules and Lucas envelopes of modules are introduced.And it is proved that the Lucas envelope Q0(I)of an ideal I is not equal to Q0(R)if and only if I is a non-semiregular ideal of R.Secondly,we intro-duce the concept of the Q0-Noetherian ring and provide the relative Hilbert Basis Theorem of the Q0-Noetherian ring,i.e.,R is a Q0-Noetherian ring if and only if the polynomial ring R[y1,y2,…,yn]is a Q0-Noetherian ring.At last,we introduce the concept of the irreducible ideal,we proved that if R is a Q0-Noetherian ring,and I is a non-semiregular ideal of R,then there are only finite minimal non-semiregular prime ideals on I.