When do the rings R[X] and R[[X]] become generalized Krull rings

Let R be a commutative ring with identity, \(\{X_{\alpha }\}\) $\{X_{\alpha }\}$ be a nonempty set of indeterminates over R, \(R[\{X_{\alpha }\}]\) $R[\{X_{\alpha }\}]$ be the polynomial ring over R, X be an indeterminate over R, and R[[X]] be the power series ring over R. In this paper, we study when \(R[\{X_{\alpha }\}]\) $R[\{X_{\alpha }\}]$ is a generalized Krull ring. We also study when R[[X]] is a generalized Krull ring, a Krull ring, a regular PIR, a Dedekind ring, a regular \(\pi \) $\pi$ -ring, or a factorial ring. Among them, we prove that R[[X]] is a generalized Krull ring if and only if R[[X]] is a Krull ring, if and only if R is a finite direct sum of Krull domains.