(Two-sided) principal ideal, generated by an element a in commutative ring is the set $\displaystyle RA=\{ra | r \in R \}$

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 $\displaystyle r_1as_1+r_2as_2+...+r_nas_n$

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 $\displaystyle r_18s_1 + r_28s_2 + r_38s_3$ 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 $\displaystyle RA=\{ra | r \in R \}$ 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