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 ?

I've never understood abstract algebra and even my number theory is kinda shit.

But as to the last question, I read in this very boring book called "the mathematician's apology" that the point of higher more "useless" math is to make the processing of more applied math easier.

A (probably somewhat stupid) example:  No one really cared about algorithms until computers came about.  Now There's an entire field devoted to that and it's all rooted in the same stuff as ever.  If/When quantum computers take off, more branches of math will need to be developed.