Let $R$ be a non commutative ring. We will say that an element of $R$ is isolated if it is zero divisor and nothing nonzero annihilates it at the same time on both sides.

Note that there are many classes of rings that do not contain isolated elements.

$\bullet$ If $R$ is an integral ring, it does not contain isolated elements.

$\bullet$ If $R$ is a nilpotent ring, it does not contain isolated elements.

$\bullet$ If $R$ is a reversible ring, it does not contain isolated elements.

$\bullet$ If $R$ is a finite unital ring, it does not contain isolated elements.

Do you know of any indecomposable ring that has no isolated elements and is neither reversible, nor integral, nor nilpotent, nor finite unital?