Weyl algebra

Think of the operations of differentiation and multiplication-by-x acting on polynomials. They almost commute, but not quite: differentiating x*f gives f plus x times the derivative, so d and x obey d*x - x*d = 1 rather than d*x = x*d. The Weyl algebra is the abstract ring generated by exactly these two operations and that one twisting relation. It is the algebra of polynomial-coefficient differential operators, distilled to its bare bones.

Precisely, the n-th Weyl algebra A_n over a field k is generated by x_1, ..., x_n and d_1, ..., d_n subject to the relations that the x's commute among themselves, the d's commute among themselves, and d_i*x_j - x_j*d_i = delta_{ij} (the Kronecker delta). It can be realized concretely as the ring of differential operators on k[x_1, ..., x_n] with polynomial coefficients, acting by multiplication and partial differentiation.

Over a field of characteristic 0 the Weyl algebra is a simple Noetherian domain that is not a division ring and is not finite-dimensional — a clean example showing that simple does not imply semisimple or Artinian. Its center is just the scalars k, it has no nontrivial finite-dimensional modules, and it has global dimension n, making it a workhorse example in homological and noncommutative algebra.

A characteristic warning: in characteristic p the picture changes drastically. There x^p and d^p become central, so A_n is no longer simple and behaves like a finite module over a large commutative center — the source of much arithmetic D-module theory. The Weyl algebra is also the canonical example where the Ore condition holds, so it embeds in a division ring of fractions despite having no finite-dimensional representations.