Non commutative ring and polynomial faggotry

Howdy, chaps ?

Say, has any one really been far even as to do look more like ?

Consider a polynom with coefficients over a non-commutative ring. How do you define roots of such a polynom ? Basicly, how do you define the induced factorisation of such a polynom ? What can you tell about the aforementionned root (numbers, relations, sauce) ?

And the least important part to us mathematicians : is this useful for anything ?

what kind of "proper" definition are you looking for, it's the _root_ of the polynomial... since it's noncommutative, you just have left and right coefficients for your polynomial

r is a root of f(x) if and only if f(r) = 0.

its harder to factor shit since you're not used to it, but you can still do it

i.e. f(x) = ax^2b-ax-xb+1 = (ax-1)(xb-1) has roots that are the right inverse of a or left inverse of b, if they exist.