Easy# Module Over a Ring: Definition

ABSALG-GC1VF5

Let $R$ be a ring (with identity) $1$.

A (unitary) left $R$-module $M$ is an abelian group $(M,+)$ together with an action of $R$ on $M$:

$R\times M\rightarrow M$, $\quad (r,m)\mapsto rm$

...such that:

$r(m+n)=rm+rn$, $\quad(r+s)m=rm+sm$, $\quad r(sm)=(rs)m$, $\quad 1m=m$

...for all $r,s\in R$ and $m,n\in M$. The definition of a right module is similar, with the action of $R$ on the right.

Very often the words "with identity" for $R$ and "unitary" for $M$ are omitted and tacitly assumed.

Which of the following $M$ is a unitary left $R$-modules?

Select **ALL** that apply.