Isonoetherian power series rings III

Abstract

Let R be a commutative unitary ring and I an ideal of R. We prove that $R+XI[[X]]$ is isonoetherian if and only if R is isonoetherian, I is idempotent and each ideal of R contained in I is finitely generated. We prove that $R[[X]]$ satisfies ACC_d on ideals if and only if R is Noetherian. We deduce that the ring $R+XI[[X]]$ satisfies ACC_d on ideals if and only if R satisfies ACC_d on ideals, I is idempotent and each ideal of R contained in I is finitely generated. We study polynomial rings satisfying ACC_d on ideals. We give an example of a ring satisfying ACC_d on ideals but is not isonoetherian.

Keywords:

• ACC_d on ideals
• isonoetherian ring
• power series ring
• SFT-ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F25
• 13E0