Nicely-contractible ideals, z0−ideals and z−ideals in formal series rings in finitely many variables

Abstract

Let A be a commutative ring with identity and $A[[X_1,\dots ,X_n]]$ the ring of formal power series in n variables. In this paper, we consider the SFT-property on A and its relation with $A[[X_1,\dots ,X_n]]$. Then, we see when the minimal prime ideals of $A[[X_1,\dots ,X_n]]$ are extended from those of A. We discuss the notion of nicely-contractible ideals in $A[[X_1,\dots ,X_n]]$ and its relation with the SFT-property. We apply our results to the $z^0-$ ideals and the $z-$ ideals of $A[[X_1,\dots ,X_n]]$.

KEYWORDS:

• Minimal prime ideal
• nicely-contractible ideal
• power series
• $z-$ ideal
• $z^0-$ ideal

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F25
• 13A15
• 13B25