  Let $R$ be a $G$-graded ring. In this article, we introduce two new concepts
on graded rings, namely, weakly graded rings and invertible graded rings, and
we discuss the relations between these concepts and several properties of
graded rings. Also, we study the concept of weakly crossed products and study
some properties defined on weakly crossed product and give the relationship
between this new concept and several properties of graded rings. Moreover, in
this article, we give a generalization for the concept of graded essential
submodules, and introduce the concept of graded semi-uniform modules which is a
generalization for the concept of graded uniform modules.
