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Module Structure of Certain Rings of Differential Operators

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Let k be a field of characteristic zero and let k((x1,...,xn)) be the field of fractions of the ring of formal power series in n variables k[[x1,..,xn]]. We denote by \(\mathscr{E}_{n}(k)\) the k-subalgebra of Endk(k((x1,...,xn))) generated by the elements of k((x1,...,xn)) and the usual derivations ∂1,...,∂n. It is shown that every left or right ideal of \(\mathscr{E}_{n}(k)\) is generated by two elements and that stably free left or right \(\mathscr{E}_{n}(k)\)-modules of rank at least 2 are free. Similar properties are proved for the ring of k-linear differential operators with coefficients in k[[x1]].

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal da Paraíba, CEP 58051-900, Cidade Universitária, João Pessoa, PB, Brazil

    N. Caro-Tuesta

  2. Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, CEP 13560-970, São Carlos, SP, Brazil

    D. Levcovitz

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  1. N. Caro-Tuesta
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  2. D. Levcovitz
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Correspondence to N. Caro-Tuesta.

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Presented by: Kenneth Goodearl

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Caro-Tuesta, N., Levcovitz, D. Module Structure of Certain Rings of Differential Operators. Algebr Represent Theor 23, 1637–1657 (2020). https://doi.org/10.1007/s10468-019-09905-4

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