# Definition of modules

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?

• 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 $R\times V\to V$. May 2, 2014 at 0:01
• 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). May 2, 2014 at 0:02
• The product of an element of a field and an element of a group is still in that group??
– anon
May 2, 2014 at 0:03
• @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. May 2, 2014 at 0:06
• @hardmath Compare my wording to OP's wording (and punctuation)...
– anon
May 2, 2014 at 0:07

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.