We prove that for a graded algebra \(\mathcal {A}\) with a derivation \(D_{\mathcal {A}}\) satisfying certain conditions, and a bi-graded algebra \(\mathcal {A}[q]\) with an extended derivation D of \(D_{\mathcal {A}}\), there are only finitely many \(D_{\mathcal {A}}\)- and D-invariant (or differential with respect to \(D_{\mathcal {A}}\) and D) principal prime ideals of \(\mathcal {A}\) and of \(\mathcal {A}[q]\), respectively. As its application, we prove the algebraic independence of the values of certain Eisenstein series for the arithmetic Hecke groups H(m) for \(m=4,6, \infty \), which is an extension of Nesterenko’s result for the Eisenstein series for SL\(_2(\mathbb {Z})\).

我们证明，对于一个分级代数\（\ mathcal {A} \）与派生\（d _ {\ mathcal {A}} \）满足一定的条件，和双渐变代数\（\ mathcal {A} [Q ] \）和\（D _ {\ mathcal {A}} \）的扩展导数D，只有有限的\（D _ {\ mathcal {A}} \\）和D不变（或相对于\（\ mathcal {A} \）和\（\ mathcal {A} [q] \）的\（D _ {\ mathcal {A}} \）和D）主要素理想， 分别。作为其应用，我们证明了\（m = 4,6，\ infty \）的算术Hecke组H（m）的某些Eisenstein级数的值的代数独立性，这是Nesterenko对Eisenstein级数的结果的扩展表示SL \（_ 2（\ mathbb {Z}）\）。