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# Finitely Generated Submodules of Differentiable Functions

B. Roth
Proceedings of the American Mathematical Society
Vol. 34, No. 2 (Aug., 1972), pp. 433-439
DOI: 10.2307/2038386
Stable URL: http://www.jstor.org/stable/2038386
Page Count: 7
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## Abstract

Suppose $\lbrack \mathscr{E}^m(\Omega) \rbrack^p$ is the Cartesian product of the space of real-valued $m$-times continuously differentiable functions on an open set $\Omega$ in $R^n$ with itself $p$-times where $m$ is finite and $\Omega$ is connected. $\lbrack \mathscr{E}^m(\Omega) \rbrack^p$ is a $\mathscr{E}^m(\Omega)$-module. The finitely generated submodules of $\lbrack \mathscr{E}^m(\Omega) \rbrack^p$ are $\operatorname{im}(F)$ where $F: \lbrack \mathscr{E}^m(\Omega) \rbrack^q \rightarrow \lbrack \mathscr{E}^m(\Omega) \rbrack^p$ is a $p \times q$ matrix $(f_{ij})_{1 \leqq i \leqq p, 1 \leqq j \leqq q}, f_{ij} \in \mathscr{E}^m(\Omega)$. In the present paper, it is shown that $\operatorname{im}(F)$ is closed in $\lbrack \mathscr{E}^m(\Omega) \rbrack^p$ if and only if the rank of the matrix $(f_{ij}(x))_{1 \leqq i \leqq p, 1 \leqq j \leqq q}$ is constant for $x \in \Omega$. Applications are made to systems of division problems for distributions.

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