Bulletin of the
Korean Mathematical Society BKMS

The aim in this note is to describe some classes of rings in relation to factorization by prime radical, upper nilradical, and Jacobson radical. We introduce the concepts of {\it tpr} ring, {\it tunr} ring, and {\it tjr} ring in the process, respectively. Their ring theoretical structures are investigated in relation to various sorts of factor rings and extensions. We also study the structure of noncommutative tpr (tunr, tjr) rings of minimal order, which can be a base of constructing examples of various ring structures. Various sorts of structures of known examples are studied in relation with the topics of this note.

Keywords: tpr ring, tunr ring, tjr ring, polynomial ring, factor ring, noncommutative ring of minimal order, nilradical, Jacobson radical

MSC numbers: 16N40, 16S36