Finitely generated submodules of differentiable functions
HTML articles powered by AMS MathViewer
- by B. Roth PDF
- Proc. Amer. Math. Soc. 34 (1972), 433-439 Request permission
Abstract:
Suppose ${[{\mathcal {E}^m}(\Omega )]^p}$ 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. ${[{\mathcal {E}^m}(\Omega )]^p}$ is a ${\mathcal {E}^m}(\Omega )$-module. The finitely generated submodules of ${[{\mathcal {E}^m}(\Omega )]^p}$ are ${\text {im}}(F)$ where $F:{[{\mathcal {E}^m}(\Omega )]^q} \to {[{\mathcal {E}^m}(\Omega )]^p}$ is a $p \times q$ matrix ${({f_{ij}})_{1 \leqq i \leqq p,}}_{1 \leqq j \leqq q},{f_{ij}} \in {\mathcal {E}^m}(\Omega )$. In the present paper, it is shown that ${\text {im}}(F)$ is closed in ${[{\mathcal {E}^m}(\Omega )]^p}$ 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.References
- B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
- B. Roth, Finitely generated ideals of differentiable functions, Trans. Amer. Math. Soc. 150 (1970), 213–225. MR 262810, DOI 10.1090/S0002-9947-1970-0262810-9
- B. Roth, Systems of division problems for distributions, Trans. Amer. Math. Soc. 155 (1971), 493–504. MR 415310, DOI 10.1090/S0002-9947-1971-0415310-8
- B. Roth, Submodules of $C(X)\times \cdots \times C(X)$, Proc. Amer. Math. Soc. 32 (1972), 543–548. MR 293382, DOI 10.1090/S0002-9939-1972-0293382-8
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 34 (1972), 433-439
- MSC: Primary 46E15
- DOI: https://doi.org/10.1090/S0002-9939-1972-0300072-1
- MathSciNet review: 0300072