An example of a ring which is very close to division ring but not a division ring.

Question

Let $R$ be a ring with unity.

1. An element $a\in R$ is said to be a unit element if there exists $b\in R$ such that $ab=ba=1$. The ring $R$ is called a division ring if every nonzero element is a unit element.
2. An element $f\in R$ is said to be a full element if there exists $r,s\in R$ such that $rfs=1$. Every unit element is a full element. If $R$ is commutative, then every full element is a unit element.

I am looking for an example of a ring $R$ whose every nonzero element is a full element but it has at least one nonzero element which is not a unit, i.e., $R$ is not a division ring.

Please suggest me something so that I can find this example.

Answer 1

Let $R$ be the ring of linear transformations of a countably infinite dimensional vector space.

It’s elementary to show that the transformations with infinite dimensional images are full elements, and they remain full when you quotient by the (unique, in this case) maximal ideal $M$ of $R$.

So now it is up to you to verify the hints.