On noncommutative piecewise Noetherian rings

ABSTRACT

We extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.

KEYWORDS:

• Associated prime ideal
• Gabriel correspondence
• Gabriel dimension
• piecewise Noetherian
• reduced rank

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16P60
• 16S50
• 16S90