Graded quotient rings

This survey is based on the PhD Thesis which was defended at the Dissertation council of the Faculty of Mechanics and Mathematics of Moscow State University on December 6, 2013. This paper is devoted to the study of quotient rings of rings graded by a group. Graded analogs of the Faith–Utumi theorem of orders of matrix rings, Goldie's theorems of orders of completely reducible rings are proved and the orthogonal graded completion, which is an analog of the quotient ring underlying the orthogonal completion theory of Beidar–Mikhalev, is constructed.