Artin-Schreier extensions and generalized associated orders

Let k be the local field Fq((T)), where q is a power of a prime number p. Let L be a totally ramified Artin-Schreier extension of degree p over k and G its Galois group, and let v be a valuation of L such that v(T)=1. Define MLr={x∈L:v(x)⩾rp}. We give a basis for the Ok-module Ar,b(L/k)={x∈k[G]:x⋅MLr⊂MLb}. Moreover, we determine the conditions for which MLr is free over the ring Ar,r.