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While perusing through the article "Algorithms for solving linear ordinary differential equations" by Winfried Fakler (a pdf can be found through a google search), I noticed Faker mentioning on page 2 that

From a mathematical point of view linear differential operators generate a left skew polynomial ring of derivation type. The elements of such a ring are called skew polynomials or Ore polynomials. For Ore polynomials the usual polynomial addition holds. Only the multiplication is different. It is declared as on extension of the rule $a\in k$ $$Da=aD+a'$$ to arbitrary Ore polynomials. Multiplication of Ore polynomials is in fact operator composition. Therefore, $k[D]$ is not a commutative ring. This means there is a left and right division. Indeed, there exists an extended Euclidean algorithm and it is possible to determine for any two nontrivial elements a smallest nontrivial common left multiple.

I'm curious to know whether anyone knows the details about this "extended Euclidean algorithm" for differential operators?

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You may find the algorithm in:

Li, Noncommutative Gröbner Bases and Filtered-Graded Transfer, Springer Lecture Notes in Mathematics, Vol. 1795. Chapter I.2

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