M-Primary Elements of a Local Noether Lattice

Abstract

In this paper, we consider the extent to which a local Noether lattice (ℒ, M) is characterized by the sub-multiplicative lattice, denoted δℒ, of M-primary elements. (Here we use the notation (ℒ, M) to indicate that M is the maximal element of ℒ.) In particular, we call ℒM-complete if, given any decreasing sequence {Ai} of elements and any n ≧ 1, it follows that Ai ≦ A V Mn for large i, where A = ΛAi And we show that, given two Mi-complete local Noether lattices (ℒ1, M1) and (ℒ2, M2), with δℒ1 ≅ δℒ2, it follows that ℒ1 ≅ ℒ2. Further, we show that any local Noether lattice (ℒ, M) is a sublattice of a local Noether lattice (ℒ*, M) which is M-complete and such that δℒ = δℒ*.