The ring A[X,Y;λ] and the SFT property

Abstract

Let A be a ring and $\lambda :{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ be an increasing nonzero function. In this paper, we show that if A is a Noetherian ring with characteristic $n\ne 0$ and $\underset{k\to +\infty }{\mathrm{\text{lim}}}\frac{\lambda (k+1)}{\lambda (k)}=+\infty$, then $A[X,Y;\lambda ]$ is an SFT ring. This result allows us to construct a nonNoetherian SFT ring which has some Noetherian completion. Also, we use the composite ring extension to construct many examples of such rings. Many examples are provided.

KEYWORDS:

• Completion
• I-adic topology
• SFT-rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F25
• 13E05
• 13J01

Acknowledgments

The authors would like to thank the referee for his/her valuable comments which helped us improve the presentation of our paper.