  In this paper, we will be delving deeper into the connection between the
rough theory and the ring theory precisely in the principle and maximal ideal.
The rough set theory has shown by Pawlak as good formal tool for modeling and
processing incomplete information in information system. The rough theory is
based on two concepts the upper approximation of a given set is the union of
all the equivalence classes, which are subsets of the set, and the lower
approximation is the union of all the equivalence classes, which are
intersection with set non-empty. Many researchers develop this theory and use
it in many areas. Here, we will apply this theorem in the one of the most
important branches of mathematics that is ring theory. We will try to find the
rough principal and maximal ideal as an extension of the notion of a principal
maximal ideal respectively in ring theory. In addition, we study the properties
of the upper and lower approximation of a principal maximal ideal. However,
some researchers use the rough theory in the group and ring theory. Our work,
is Shaw there are rough maximal and principle ideal as an extension of the
maximal and principle ideal respectively. Our result will introduce the rough
maximal ideal as an extended notion of a classic maximal ideal and we study
some properties of the lower and the upper approximations a maximal ideal.
