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.
Citation
V. V. BAVULA. "Classifications of Prime Ideals and Simple Modules of the Weyl Algebra $A_1$ in Prime Characteristic." Tokyo J. Math. 46 (1) 161 - 191, June 2023. https://doi.org/10.3836/tjm/1502179377
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