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

Let A be a ring and λ:R≥0→R≥0 be an increasing nonzero function. In this paper, we show that if A is a Noetherian ring with characteristic n≠0 and limk→+∞λ(k+1)λ(k)=+∞, then AX,Y;λ 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.