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Let $q_1, q_2, q_3 \in \mathbb{Z}[x,y]$ such that $q_1, q_2$ are algebraically independent and let $S$ be an algebra generated by $q_1, q_2, q_3$ over $F_p$. If writing $S$ as a module over $F_p [q_1, q_2]$ gives $S =F_p [q_1, q_2] \oplus F_p [q_1, q_2] q_3$, then can we say that the algebra $A$ which is generated by $q_1, q_2, q_3$ over $\mathbb{Z}$ can be written, as a module over $\mathbb{Z} [q_1, q_2]$, as $A =\mathbb{Z} [q_1, q_2] \oplus \mathbb{Z} [q_1, q_2] q_3$? If we have the $p$-adic integers instead of integers, does it give the same answer?

Thank you

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  • $\begingroup$ The answer is almost certainly "no", but the least you should do is specify quantifiers in your question. $\endgroup$
    – Alex Degtyarev
    Commented Jul 9, 2015 at 15:56
  • $\begingroup$ @AlexDegtyarev So, in general it is no. How about $p$-adic integers? $\endgroup$
    – user75877
    Commented Jul 9, 2015 at 16:05
  • $\begingroup$ @AlexDegtyarev If the coefficients of $q_i$ aren't a multiple of $p$, is the statement correct? $\endgroup$
    – user75877
    Commented Jul 9, 2015 at 16:12

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