High Quality Content by WIKIPEDIA articles! In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), f_n(X) partial_X^n + cdots + f_1(X) partial_X + f_0(X). More precisely, let F be a field, and let F(X) be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F(X). ?X is the derivative with respect to X. The algebra is generated by X and ?X. The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. The Weyl algebra is a quotient of the free algebra on two generators, X and Y, by the ideal generated by the single relation: YX - XY - 1.

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