(Two-sided) principal ideal, generated by an element a in commutative ring is the set
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
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 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 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