Convergence to the equilibrium for the Pauli equation without detailed balance condition

Let $A?B$ be an extension of commutative rings with identity, X an analytic indeterminate over B, and $R\text{:}=A+XBX]]$, the subring of the formal power series ring $BX]]$, consisting of the series with constant terms in A. In this Note we study when the ring R is Noetherian. We prove that R is Noetherian if and only if A is Noetherian and B is a finitely generated A-module. To cite this article: S. Hizem, A. Benhissi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).