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 ?

>>9
i have no idea about number of roots of a polynomial over non-commutative ring, maybe one of these books might have something.

A First Course in Noncommutative Rings by T.Y. Lam
http://ifile.it/pdi5017/lam.djvu

Noncommutative Algebra by Benson Farb, R. Keith Dennis
http://www.mediafire.com/?mm4zm3eemq3

wikipedia says there are left/right euclidean algorithms over these rings. but you should remember that polynomials dont always factor completely into their roots, just irreducible factors (i'm guessing left/right irreducible in non-commutative case), and the factorization might not be unique.

also see
www.emis.de/journals/GMJ/vol10/v10n4-10.ps+roots+of+a+noncommutative+polynomial&cd=12&hl=en&ct=clnk&;gl=us" target='_blank'>http://74.125.155.132/search?q=cache:CVVS3xHTxGAJ:www.emis.de/journals/GMJ/vol10/v10n4-10.ps+roots+of+a+noncommutative+polynomial&cd=12&hl=en&ct=clnk&;gl=us
http://en.wikipedia.org/wiki/Noncommutative_fundamental_theorem_of_algebra

dont know if those will be helpful

>>10
i think you meant L = LGR = R, but yeah, i guess so