Some results on finitely Laskerian rings

摘要

In Section 1 of this note we give an example of a strongly Laskerian domain D for which the polynomial ring Dx admits a 2-generated ideal which does not admit a primary decomposition. In Section 2 of this note we prove that if R is a quasilocal ring with M as its unique maximal ideal such that R/AnnM is Artinian, then for any subring T of the polynomial ring Rx, each finitely generated proper ideal of T admits a primary decomposition.