The research field of graded ring theory is a rich area of mathematics with many connections to e.g. the field of operator algebras. In the last 15 years, algebraists and operator algebraists have defined algebraic analogues of important operator algebras. Some of those analogues are rings that come equipped with a group grading. We want to reach a better understanding of the graded structure of those analogue rings. Among group graded rings, the strongly graded rings stand out as being especially well-behaved. The development of the general theory of strongly graded rings was initiated by Dade in the 1980s and since then numerous structural results have been established for strongly graded rings.

 In this thesis, we study the class of epsilon-strongly graded rings which was recently introduced by Nystedt, Öinert and Pinedo. This class is a natural generalization of the well-studied class of unital strongly graded rings. Our aim is to lay the foundation for a general theory of epsilon-strongly graded rings generalizing the theory of strongly graded rings. This thesis is based on three articles. The first two articles mainly concern structural properties of epsilon-strongly graded rings. In the first article, we investigate a functorial construction called the induced quotient group grading. In the second article, using results from the first article, we generalize the Hilbert Basis Theorem for strongly graded rings to epsilon-strongly graded rings and apply it to Leavitt path algebras.  In the third article, we study the graded structure of algebraic Cuntz-Pimsner rings. In particular, we obtain a partial classification of unital strongly, epsilon-strongly and nearly epsilon-strongly graded Cuntz-Pimsner rings up to graded isomorphism.

The development of a general theory of strongly group graded rings was initiated by Dade, Năstăsescu and Van Oystaeyen in the 1980s, and since then numerous structural results have been established.  In this thesis we develop a general theory of so-called (nearly) epsilon-strongly group graded rings which were recently introduced by Nystedt, Öinert and Pinedo and which generalize strongly group graded rings. Moreover, we obtain applications to  Leavitt path algebras, unital partial crossed products and algebraic Cuntz-Pimsner rings. 

This thesis is based on five scientific papers (A, B, C, D, E). 

Papers A and B are concerned with structural properties of epsilon-strongly graded rings. In Paper A, we consider an important construction called the induced quotient group grading. In Paper B, using results from Paper A, we obtain a Hilbert Basis Theorem for epsilon-strongly graded rings.  In Paper C, we study the graded structure of algebraic  Cuntz-Pimsner rings. In particular, we obtain a partial characterization of unital strongly graded, epsilon-strongly graded and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. 

In Paper D, we give a complete characterization of group graded rings that are graded von Neumann regular.

Finally, in Paper E, written in collaboration with Lundström, Öinert and Wagner, we consider prime nearly epsilon-strongly graded rings. Generalizing Passman's work from the 1980s, we give  necessary and sufficient conditions for a nearly epsilon-strongly graded ring to be prime.