3-standardness of the maximal ideal

We study a notion called n-standardness (defined by M.E. Rossi (2000) in [10], and extended in this paper) of ideals primary to the maximal ideal in a Cohen–Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is 3-standard, first proving results for when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic 0 case. As an application, we extend a result due to T. Puthenpurakal (2005) [9] and show that a certain length associated with a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.