Abstract
The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w-Noetherian domains. Let is a commutative ring and
is a divided prime ideal of
Let
be a ring with total quotient ring T(R), and set
such that
for every
and every
Then
is a ring homomorphism from T(R) into
and
restricted to R is also a ring homomorphism from R into
given by
for every
At this setting, we introduce the notions of
-piecewise Noetherian rings and
-piecewise w-Noetherian rings and we show that the theories of
-piecewise Noetherian (resp.
-piecewise w-Noetherian) rings resemble those of piecewise Noetherian (resp. piecewise w-Noetherian) domains.
2010 Mathematics Subject Classification:
Acknowledgements
The authors would like to express their sincere thanks for the referee for his/her helpful suggestions and comments, which have greatly improved this paper.