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'm pretty sure left/right inverses are the same in associative algebras.

Lemma: identity element is unique (if there exists a left and right identity)
Suppose there are left/right identities: L,R, respectively.
Then L = LR = R by the definition of the corresponding identities.  Thus, L = R, and there is only 1 identity.

Suppose there are left/right inverses of an element g: l,r, respectively.  Then
rg = Identity(rg) = (lg)(rg) = l(gr)g = lg = Identity
Thus, right inverses are also left inverses.