Modules are known to generalize abelian groups since all abelian groups are $\mathbb{Z}$-module (scalar multiplication : $n\cdot x=x+x+\dots +x$ (n times)) However, all modules are abelian group, when not considering scalar multiplication. Therefore it's hard to understand that module is more generalized concept than abelian group. What I mean is, definition of module is much more specific than that of abelian group(it fully contains it). What is the real difference between abelian groups and modules? Can you provide some intuition in understanding?
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2$\begingroup$ Well, it's what you said: the difference is having the action of an arbitrary ring by scalar multiplication. $\endgroup$– Sassatelli GiulioCommented Aug 14 at 0:13
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1$\begingroup$ Not all abelian groups can be modules over the same rings in nontrivial ways. That difference is often useful. $\endgroup$– RandallCommented Aug 14 at 1:19
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$\begingroup$ In some sense, as do nilpotency and solvability. $\endgroup$– czech-spanierCommented Aug 14 at 3:59
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The way in which an $R$-module generalizes an abelian group is that the definition of an $R$-module consists of axioms (such as $r(m+n) = rm + rn$) that are true statements for abelian groups with respect to scalar multiplication by $\mathbb{Z}$.
Your point that $R$-modules do not really generalize abelian groups does not hold water. There is indeed an underlying abelian group structure for every $R$-module. However, that is obtained by forgetting the scalar multiplication by $R$. Hence, the $R$-module structure is lost; e.g., the above axiom cannot be stated.
