This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions a coefficient by (elementary) matrix-variables over ground ring in not necessary commuting variables. These subalgebras full rings show up noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining precise lift Hilbert Basis Theorem when...

We extend the notion of finite embedding problems over fields, a central in inverse Galois theory, to situation centrally division rings. first show that solving problem such ring H is equivalent finding solution some center involving polynomial constraint. then every constant split skew field rational fractions H(t) with indeterminate t has solution, if an ample field, thus providing non-commu...

Hodges and Levasseur described the primitive spectra of quantized coordinate rings of SL3 [14], and then of SLn [15]. These results established a close connection between primitive ideals, torus action, and Poisson geometry. The proofs relied on explicit computations involving generators and relations. Subsequently, Joseph generalized the Hodges–Levasseur program to semisimple algebraic quantum...

The aim of this work is to study some noncommutative differential operators. We take an algorithmic approach as well as further developing the mathematics, and design and analyse algorithms to solve fundamental problems. We also give applications to differential equations. First we consider how to factor skew polynomials. These are polynomials in a differential or difference operator. Using the...