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This paper consists of three parts: (I) To develop general theory of a(large) class of central simple finite dimensional algebras and answering somenatural questions about them (that in general situation it is not even clearhow to approach them, and the Brauer group is a step in the right directions),(II) To introduce and develop general theory of a large class of rings,PLM-rings, (intuitively, they are the most general form of Quillen' Lemma), and(III) to apply these results for the following four classic algebras thatturned out to be PLM-rings.  Let $K$ be an arbitrary field and $q\in K\backslash \{ 0,1\}$ be a primitive$n$'th root of unity. Classifications of prime, completely prime, maximal andprimitive ideals, and simple modules are obtained for the quantum Weyl algebra$A_1(q)=K\langle x,y \, | \, xy-qyx =1\rangle$, the skew polynomial algebra$\mathbb{A} = K[h][x;\sigma]$, the skew Laurent polynomial algebras$\mathcal{A} := K[h][x^{\pm 1};\sigma ]$, and $\mathcal{B} := K[h^{\pm1}][x^{\pm 1};\sigma ]$ where $\sigma (h) = qh$. The quotient rings (offractions) of prime factor algebras of the algebras $A_1$, $\mathbb{A}$,$\mathcal{A}$, and $\mathcal{B}$ are explicitly described. Each quotient ringis a central simple finite dimensional algebra, i.e., isomorphic to the matrixalgebra $M_d(D)$ for some $d\geq 1$ and a central simple division algebra $D$.The division algebra $D$ is either a finite field extension of $K$ or a {\emcyclic} algebra. These descriptions are a key fact in the classifications ofprime ideals, completely prime ideals, and simple modules for the algebrasabove. For each simple module its basis, dimension, and endomorphism algebraare given. Explicit descriptions are obtained for the automorphism groups ofthe algebras $A_1$, $\mathbb{A}$, $\mathcal{A}$, and $\mathcal{B}$.

Classifications of prime ideals and simple modules of the quantum Weyl algebra $A_1(q)$ ($q$ is a root of unity)