Noetherian property of subrings of power series rings II

Let R be a commutative ring with unit. We study certain subrings R[X; Y, lambda] of R[X][[Y] = R[X-1, ..., X-n][[Y-1, ...,Y-m,]] where A is a nonnegative real-valued increasing function. These subrings naturally arise from studying p-adic analytic variation of zeta functions over finite fields. In our previous work, we gave a necessary and sufficient condition for R[X; Y, lambda] to be Noetherian when Y has more than one variable and lambda grows as fast as linear. In this paper, we show that the same result holds even when Y has only one variable. This contradicts Davis and Wan's result stating that R[X; Y, lambda] is always Noetherian if R is a field. We however found a mistake in their proof. (C) 2015 Elsevier B.V. All rights reserved.