Thread: Question about principal ideals in noncommutative rings

1. [SOLVED] Question about principal ideals in noncommutative rings

(Two-sided) principal ideal, generated by an element a in commutative ring is the set $RA=\{ra | r \in R \}$
Actually this definition results from the more general one, which does not assume the ring is commutative.
It says that (two-sided) principal ideal, generated by element a is the set of all finite sums of type $r_1as_1+r_2as_2+...+r_nas_n$

I could easily generate example to convince myself that the second definition is equivalent to the first in commutative rings. For example taking the ring to be Z, taking a to be, say 8, and trying various finite sums of type $r_18s_1 + r_28s_2 + r_38s_3$ where r and s are elements of Z is convincing enough for me.

But now I'm lazy (and scared that it could take the better part of workday) if I try myself to generate an example where the set $RA=\{ra | r \in R \}$ is not closed under addition in noncommutative ring. I was thinking to try with matrix rings (in which elements from some finite field) to minimize computations). Any shortcuts and ready examples will be appreciated

2. Re: Question about principal ideals in noncommutative rings

I did an example with 2x2 matrices with elements from Z2. It turned out to be easy.