  In this work, we study several equivalence relations induced from the
partitions of the sets of words of finite length. We have results on words over
finite fields extending the work of Bacher (2002, Europ. J. Combinatorics, {\bf
23}, 141-147). Cardinalities of its equivalence classes and explicit
relationships between two words are determined. Moreover, we deal with words of
finite length over the ring $\mathbb{Z}/N\mathbb{Z}$ where $N$ is a positive
integer. We have arithmetic results parallel to Bacher's.
