Algebraic properties of separated power series (Q926251)

Separated power series rings were introduced by \textit{R. Cluckers, L. Lipshitz} and \textit{Z. Robinson} [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 4, 535--568 (2006; Zbl 1168.12006)] as part of an effort to develop analytic motivic integration theory. This paper has already established commutative algebraic properties of these rings like the Weierstrass Division Theorem. The present paper continues this effort by establishing commutative algebraic properties of separated power series rings with parameters from a field, which were also introduced in the above paper. In particular, the author proves the Weierstrass Division Theorem, the Weierstrass Preparation Theorem, the Nullstellensatz and the Noether Normalization Theorem for these rings. It is concluded that these rings are Cohen--Macaulay, universally caternary and unique factorization domains. It is hoped that these results will enable the use of algebraic geometric methods in the analytic geometry over non-Archimedean valued fields.