the algebra of strictly convergent power series over $k$: $T_n = \{ \sum_{\mu} a_\mu X^\mu : |a_\mu| \stackrel{|\mu| \rightarrow \infty}{\rightarrow}  0 \}$. The algebra, $T_n$, is a noetherian and factorial ring with many useful properties, and it is the basis for studying $k$-affinoid algebras, see Bosch etc [@Bosch]. A $k$-algebra, $A$, is called $k$-affinoid if there exists a continuous epimorphism, $T_n \rightarrow A$, for some $n \geq 0$. Given $\rho = (\rho_1, \cdots, \rho_n)$ in ${\mathbf{R}}^n$, where $\rho_i > 0$ for each $i$, one can define $$\begin{aligned}
T_{n}(\rho) &=& \{ \sum_{\mu} a_\mu X^{\mu} \in k[[X_1, \cdots, X_n]] : |a_\mu|\rho_1^{\mu_1}\cdots \rho_n^{\mu_n} \stackrel{|\mu| \rightarrow \infty} {\rightarrow} 0\}.\end{aligned}$$ Note that $T_n(1,\cdots, 1) = T_n$. Furthermore, $T_n(\rho)$ is $k$-affinoid if, and only if, $\rho_i$ is an element of $|k_a^\ast|$ for all $i$, where $k_a$ is the algebraic closure of $k$, from which one can immediately verify that it is noetherian. It is shown by van der Put in [@Put] that this ring is noetherian for any $\rho$ in ${\mathbf{R}}^n$, where $\rho_i > 0$ for each $i$. Define the Washnitzer algebra $W_n$ to be $$\begin{aligned}
 W_n &=& \bigcup_{\rho\in {\mathbf{R}}^n, \rho_i > 1} T_n(\rho).\end{aligned}$$ It is shown in Güntzer [@G] that $W_n$ is noetherian and factorial. 