Paragraded rings and their ideals

The notions of a paragraded ring and a homogeneous ideal, which are at the same time a generalization of the classical graduation, as defined by Bourbaki, and an extension of the earlier work done by M. Krasner, were introduced by M. Krasner and M. Vuković. After recalling the notion of paragraded rings, we list out and prove several facts about them. One of the most important properties is that the homogeneous part of the direct product and the direct sum of paragraded rings are the direct product and the direct sum of the corresponding homogeneous parts, respectively. Next we give the notion of a homogeneous ideal of a paragraded ring and prove that the factor ring obtained from a paragraded ring and its homogeneous ideal is also a paragraded ring. After that, we deal with basic facts about homogeneous ideals.