Factorization in Quantum Planes

Abstract

These results stem from a course on ring theory. Quantum planes are rings in two variables x and y such that yx = qxy where q is a nonzero constant. When q = 1 a quantum plane is simply a commutative polynomial ring in two variables. Otherwise a quantum plane is a noncommutative ring.Our main interest is in quadratic forms belonging to a quantum plane. We provide necessary and sufficient conditions for quadratic forms to be irreducible. We find prime quadratic forms and consider more general polynomials. Every prime polynomial is irreducible and either central or a scalar multiple of x or of y. Thus there can only be primes of degree 2 or more when q is a root of unity.

Normal Quadratics in Ore Extensions, Quantum Planes, and Quantized Weyl Algebras

“…Similar results on quantum planes (which are extended here) were completed by the faculty coauthor and another undergraduate student (see [4]). Mrs. Holtz assisted with the results in Sect.…”

“…We must prove that if q = −1 and F = ax 2 + cy 2 + f is irreducible in O −1 (k 2 ) then F is prime. If a = 0 or c = 0 then this follows from [4,Theorem 13]. We easily pass to the case c = 1.…”

“…Homogeneous irreducible quadratic forms in O q (k 2 ) were classified in [4]. Theorem 6 extends this slightly.…”

“…Remark 4 and Part 3 of Lemma 2 implies that an irreducible quadratic form of this type is normal if and only if d = 0 and q = −1. This turns out to be the most interesting case.The possible prime polynomials in a quantum plane are limited by[4, Theorem 12]. If q is not a root of unity, then a prime polynomial in O q (k 2 ) must be a scalar multiple of x or of y.…”

Irreducible skew polynomials over domains

“…However, most of the results on the irreducibility of skew polynomials in R obtained so far assume that S is a division algebra. Some first results on factoring certain skew polynomials of degree two in quantum planes and quantized Weyl algebras were collected in [8,11,7].…”