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The definition of modules confuses me:

$R$ is a ring, then a left $R$ module is an abelian group $V$ together with a multiplication map $$R \times V \to V, (r,v) \to rv$$ satisfies some natural axioms.

So from the definition,is that means $rv \in V$ ? But since $R$ is a ring and $V$is an abelian group, so the product of an element of ring and an element of group is still in that group ?? I know the vector space example, and it is easy to understand that a vector multiply a scalar is still a vector, so how do you explain module?

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  • $\begingroup$ It's the same as with vector spaces... It's a "vector" (one wouldn't use that word here) by definition of the multiplication map $R\times V\to V$. $\endgroup$ May 2, 2014 at 0:01
  • $\begingroup$ A module is just like a vector space in this regard. Multiplying by an element $r$ of the ring $R$ maps a module element $v$ to module element $rv$. Of course that is just notational shorthand. An abelian group is the same as a $\mathbb{Z}$-module if we let integers "multiply" group elements in the obvious way (repeated addition). $\endgroup$
    – hardmath
    May 2, 2014 at 0:02
  • $\begingroup$ The product of an element of a field and an element of a group is still in that group?? $\endgroup$
    – anon
    May 2, 2014 at 0:03
  • $\begingroup$ @seaturtles: A vector space is an abelian group under addition, so if you have defined vector space operations, you have defined a special case of module, where the ring is a field. $\endgroup$
    – hardmath
    May 2, 2014 at 0:06
  • $\begingroup$ @hardmath Compare my wording to OP's wording (and punctuation)... $\endgroup$
    – anon
    May 2, 2014 at 0:07

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There is an operation $\cdot : R\times V \rightarrow V$. We normally suppress the dot when it is understood that an expression is a product of an element of $R$ (on the left) and an element of $V$ (on the right). It's just some map with a bunch of properties (which you mention). This is very similar to a ring "just" being an abelian additive group with an additional operation.

In vector spaces, you have a similar operation receiving a scalar and a vector and producing a vector. It's just more familiar, so you don't think about it (or you think about it entirely in components). We even use the language "scalar multiplication" and "vector multiplication" to make this distinction (and "scalar-vector multiplication" when other vector products are floating around and/or we're being pedantic).

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