The usual commutative polynomial ring in several variables over a commu- tative ring R, R[x1, . . . , xn], can be generalized to a noncommutative context by changing R to any noncommutative ring, but preserving the basic rules of multiplication, i.e., xixj = xjxi and rxi = xir for 1 ≤ i,j ≤ n and any r ∈ R. However, in some areas of mathematics and its applications, such as algebras of differential operators in differential equations and linear multidimensional control systems in algebraic analysis (see [91], [92], [93], [95], [316], [318], [319], [320], [317] and [323]), it is necessary to consider wider classes of rings of polynomial type in which the variables do not commute or the coefficients do not commute with the variables. For example, take a homogeneous linear ordinary differential equation with coefficients in Q[t],
pn(t)y(n) +···+p1(t)y′ +p0(t)y=0, pi(t)∈Q[t], 0≤i≤n;