Rings with divisibility on descending chains of ideals

Abstract

This paper deals with the rings which satisfy DCCd condition. This notion has been introduced recently by R. Dastanpour and A. Ghorbani (2017) as a generalization of Artnian rings. It is of interest to investigate more deeply this class of rings. This study focuses on commutative case. In this vein, we present this work in which we examine the transfer of these rings to the trivial, amalgamation and polynomial ring extensions. We also investigate the relationship between this class of rings and the well known ones. Furthermore, many new results are presented in the scope of this paper. For example, there is one which concerns the decomposition of ideals on prime ones and another which investigate the Krull dimension of the ring satisfying DCCd condition. At the end of this work, we provide a result which concerns the modules over such rings. Keywords: DCCd; amalgamation of ring; trivial ring extension; Noetherian ring; Artinian ring; polynomial ring extension