# A difficulty in understanding an example of a module.

My professor wrote: "If $R$ is any ring, $A$ is any abelian group and we define $ra = 0$ $\forall$ r $\in$ $R$ $\forall$ a $\in$ $A$, then $A$ will be a left $R$-module " but this statement is not clear for me, will this be the trivial module {0}, is my understanding right?

• Your professor's statement is not true with the usual definition of module, which required that the multiplicative unit of $R$ act as the identity. – Qiaochu Yuan Sep 27 '17 at 8:23
• @QiaochuYuan Your statement assumes that $R$ has a unit. That's not necessarily true (at least I think it's not). – Arthur Sep 27 '17 at 8:44
• @QiaochuYuan I think my ring is not with unit or it can not be with unit because of the given condition. – Intuition Sep 27 '17 at 9:57
• It's a bad convention to have "ring" mean "not-necessarily-unital ring" without any explanation. Most mathematicians don't use this convention so it's just confusing. – Qiaochu Yuan Sep 27 '17 at 17:57

No. The trivial module has a single element (usually denoted by $0$). Besides that, I don't understand what's not clear about the statement. All module axioms are satisfied, right?!
• yes all axioms are satisfied....but the given condition leads to that the ring action on the elements of $A$ always leads to 0....right? what is the importance of this type of modules? – Intuition Sep 27 '17 at 9:54