The following is taken from "Modules an approach to linear algebra" by Blyth
$\color{Green}{Background:}$
$\textbf{Exercise:}$
An $R$-module $M$ is said to be $\textit{simple}$ if it has no submodules other than $M$ and $\{0\}.$ Prove that $M$ is simple if and only if $M$ is generated by every non-zero $x\in M.$
$\color{Red}{Questions:}$
Should the beginning phrasing of the question be "[A]n unitary $R$-module $M$ is said to be simple..." instead of just "[A]n $R$-module $M$ is said to be simple...." The point when simple modules are discussed, must it contain the identity element? Sometimes I just see "A simple module...." instead of A unitary simple module...." Specifically in cases where one is asked to show that $M$ is a simple module in $R$ if and only if it satisfies certain condition. So assuming that it satisfies certain condition, then to show that $M$ is a simple module in $R$ requires one to show that $1_Rx=x$ for all $x\in M$. I am sorry if I am asking something obvious. I just want to make sure my gut feeling about this definition is correct.
Thank you in advance.
Contain? No, that doesn't make any sense. "unitary right $R$ module" means that $m\cdot 1_R=m$ for all $m\in M$, where $1_R$ is the identity of $R$.
Yes. In very rare occasions they might not, but in such case the author would be explicit about it.
If you really are interested in non-unitary modules, you can look at this:
If $M$ is a (possibly nonunitary) right module over a ring $R$ with identity $1$, then $M= M\cdot 1\oplus \mathrm{Ann}_M(1)$, where $M\cdot 1:=\{m\cdot1\mid m\in M\}$ is clearly a unitary $R$ module.