Abstract: Let R be a finite principal left ideal ring. Place a lexicographic ordering on the vectors in the free left R-module R^n with respect to some basis B. The code produced by searching through this list, greedily collecting vectors satisfying some property P, is called a lexicode. For decades, several iterations of a greedy algorithm have appeared, to produce maximal lexicodes with nice properties. The most recent, in 2014, was generalized to work over finite chain rings. In this talk, I will present a new version of the greedy algorithm, generalized to produce lexicodes over a much larger class of rings.