# Journal of Operator Theory

Volume 41, Issue 1, Winter 1999 pp. 127-138.

Algebraic reduction for Hardy submodules over polydisk algebras**Authors**: Kunyu Guo

**Author institution:**Department of Mathematics, Fudan University, Shanghai, 200433, P.R. China

**Summary:**For a Hardy submodule $M$ of $H^2({\bbb D}^n)$, assume that $M\cap\cal C$ (or $M \cap{\cal R}$) is dense in $M$, where $\cal C$ (or ${\cal R}$) is the ring of all polynomials (or ${\cal R}$ is a Noetherian subring of $\Hol({\overline{{\bbb D}}}^n)$ contai ning $\cal C$). We describe those finite codimensional submodules of $M$ by considering zero var ieties. The codimension formulas related to zero varieties, and some algebraic reduction theorems are obtained. These results can be regarded as generalizations of the result of Ahern-Clark ([2]). Finally, we point out that the results in this paper extend with essentially n o change to any reproducing Hilbert $A(\Omega)$-module $H$ which satisfies certain techn ical hypotheses.

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