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.
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 R/bR)
Problem I'm having is trying to show
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!....