Let \(r_1,\dots,r_n>0\) with the possibility that some \(r_i\) are not in \(|K^\times|\). The generalized Tate algebra is defined as
\[ T_{\overline{r}}=k\langle r_1^{-1}X_1,\dots,r_n^{-1}X_n\rangle=\left\{\sum_{\alpha\in\mathbb{N}^n}a_\alpha X^\alpha\mid a_\alpha\in K,\ \lim_{|\alpha|\to\infty}|a_\alpha|r^\alpha=0\right\}. \]
The norm on \(T_{\overline{r}}\) is given by
\[ \left|\sum_{\alpha}a_\alpha X^\alpha\right|_{\overline{r}}=\max_\alpha |a_\alpha|r^\alpha. \]
This is in fact a multiplicative norm and it gives a \(K\)-Banach algebra structure on \(T_{\overline{r}}(K)\).
These are a little bit more subtle to work with than the usual Tate algebras, since we don’t have a Weierstrass division theorem in general. However, they still share many nice properties with the usual Tate algebras, such as being Noetherian, factorial, and Jacobson.
Example. Let \(r\notin |K^\times|\), then consider \(K\langle r^{-1}X, rX^{-1}\rangle\).
This can be identified with \(k\langle r^{-1}X, rY\rangle/(XY-1)\) and the strange thing about the algebra is that it is a field, with \(\left| \right|_{r,\frac{1}{r}}\) being an absolute value. In principle we would expect the dimension of \(X\) to be \(1\), but we could not get this dimension from the usual Krull dimension.