Let (A,m) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with l(0 :m I)<∞, then we show that for a minimal reduction J of I,(0 : m JI)=...Let (A,m) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with l(0 :m I)<∞, then we show that for a minimal reduction J of I,(0 : m JI)=(0 :m I^2) if and only if l(0:M I^n+1)=l(0:m J)^(n+d/d)-l(0 :M J)/(0 :M I))(n+d-1/d-1) for all n≥> 0. Moreover, we study the dual of Burch's inequality. In particular, the Burch's inequality becomes an equality if G(I,M) is co-Cohen-Macaulay.展开更多
文摘Let (A,m) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with l(0 :m I)<∞, then we show that for a minimal reduction J of I,(0 : m JI)=(0 :m I^2) if and only if l(0:M I^n+1)=l(0:m J)^(n+d/d)-l(0 :M J)/(0 :M I))(n+d-1/d-1) for all n≥> 0. Moreover, we study the dual of Burch's inequality. In particular, the Burch's inequality becomes an equality if G(I,M) is co-Cohen-Macaulay.
