As we know, if $M$ is an irreducible R-module, then $End_R(M)$ has the property that any nonzero element of it is invertible.
what about the reverse？
if $End_R(M)$ has the property that any nonzero element of it is invertible, is M irreducible,or not?
Any idea will be helpful.
It’s well-known that the converse is not true. I’m sure this is mentioned in many noncommutative algebra texts. Perhaps it is a matter of using different terminology. You should definitely prefer “invertible” to “reversible”, and know that “irreducible” is often called “simple” nowadays.
See for example Gangyong Lee, Cosmin S. Roman & Xiaoxiang Zhang (2014) Modules Whose Endomorphism Rings are Division Rings, Communications in Algebra, 42:12, 5205-5223, DOI: 10.1080/00927872.2013.836211