Classifications of Prime Ideals and Simple Modules of the Weyl Algebra $A_1$ in Prime Characteristic

Abstract

Let $K$ be an arbitrary field of characteristic $p>0$. Classifications of prime ideals and simple modules are obtained for the Weyl algebra $A_1=K\langle x,\partial \, : \, \partial x-x\partial =1\rangle$, the skew polynomial algebra $\mathbb{A} = K[h][x;\sigma ]$ and the skew Laurent polynomial algebra $\cal{A} := K[h][x^{\pm 1};\sigma ]$ where $\sigma (h) = h-1$. In particular, classifications of prime, completely prime, maximal and primitive ideals are obtained for the above algebras. The quotient ring (of fractions) of each prime factor algebra of $A_1$, $\mathbb{A}$ and $\cal{A}$ is described. It is either a matrix algebra over a field or else a cyclic algebra. These descriptions are a key fact in the classification of completely prime ideals and simple modules for the algebras above.