Hi guys,

I'm trying to show that the free associative algebra on x and y over Z, Z<x,y> is not a UFD.

I start by considering the element xyx + 2x.

Then

xyx+2x = x(yx+2)=(xy+2)x

and so I need to show that yx+2 is not similar to xy+2

(a is similar to b iff R/aR $\displaystyle \cong$ R/bR)

Problem I'm having is trying to show

$\displaystyle R/(yx+2) \ncong R/(xy+2)$

For groups, to show a mapping is not an isomorphism I generally look at the order of elements or some invariant property, but I'm not too comfortable with mappings between quotient rings.

Any help at all is appreciated!....