How to formally prove that if $R={0}$, the only R-module is the zero module?

Question

I think there is something related to the axiom for an R-module $V$, $1v=v$. And for a zero ring since $1=0$ we have $0v=1v$. Hence there maybe a contradiction. But so far, I am not able to find it out.

Am I on the right track of proving it formally? Could someone please give a full prove of that?

Answer 1

You can show that $0_Rv = 0_M$ for every $v \in M$. (Just note $0_R v = (0_R + 0_R)v = 0_Rv + 0_RV$ and thus $0_M = 0_R v$.) Since $1_R v = v$, it follows in case $1_R = 0_R$ that $v = 1_R v = 0_R v = 0_M$ for every $v$, whence the claim.