Editing Differential operator (section)

===Ring of multivariate polynomial differential operators===

If ''R'' is a ring, let <math>R\langle D_1,\ldots,D_n,X_1,\ldots,X_n\rangle</math> be the non-commutative polynomial ring over ''R'' in the variables <math>D_1,\ldots,D_n,X_1,\ldots,X_n</math>, and ''I'' the two-sided ideal generated by the elements 
:<math>(D_i X_j-X_j D_i)-\delta_{i,j},\ \ \ D_i D_j -D_j D_i,\ \ \ X_i X_j - X_j X_i</math>
for all <math>1 \le i,j \le n,</math> where <math>\delta</math> is [[Kronecker delta]]. Then the ring of multivariate polynomial differential operators over ''R'' is the quotient ring {{nowrap|<math>R\langle D_1,\ldots,D_n,X_1,\ldots,X_n\rangle/I</math>.}}

This is a {{nowrap|non-commutative}} [[simple ring]].
Every element can be written in a unique way as a ''R''-linear combination of monomials of the form {{nowrap|<math>X_1^{a_1} \ldots X_n^{a_n} D_1^{b_1} \ldots D_n^{b_n}</math>.}}