I want to clarify one concept. Let $M$ be a left-module over a ring $R$.
The cyclic submodule generated by $a\in M$ is $\langle a\rangle=\{ca\mid c\in R\}$. So is $a+a$ always in such submodule? From the definition, I can't see the reason.
2026-04-01 19:43:57.1775072637
Is $a+a$ contained in the cyclic submodule over $R$?
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Well, suppose the ring $R$ has a unit element $1$. Then the ring contains $2=1+1$, $3=2+1=1+1+1$ and so on. Thus $2a = (1+1)a=a+a\in\langle a\rangle$.
The definition of $R$-module usually requires $R$ to have a unit element. If not, you can always argue that a submodule must be closed under addition.