TITLE: About the definition of a module.
QUESTION [5 upvotes]: The following is the definition of a module from the Wikipedia.
Suppose that $R$ is a ring and $1R$ is its multiplicative identity. A left $R$-module $M$ consists of an abelian group $(M, +)$ and an operation $R × M → M$ such that for all $r, s$ in $R$ and $x, y$ in $M$, we have:
$r(x+y) = rx + ry$
$(r+s)x = rx + sx$
$(rs)x = r(sx)$ 
$1_Rx = x$. 
I thought the additions in $R$ and in $M$ are different rigorously and for the first and second conditions these two additions are used without distinction. Am I right?

REPLY [3 votes]: You're right: The $+$ in each line means something different. To be extremely careful, if we have the group $(M, +)$ and the ring $(R, \oplus, \odot)$, and an operation $\star : R \times M \to M$, then the axioms should read
$$r \star (x + y) = r \star x + r \star y$$
and
$$(r \oplus s) \star x = r\star x + s \star x$$
respectively. Likewise, the third axiom is $$ (r \odot s) \star x = r \star (s \star x)$$
Of course, writing it this way is rather cumbersome, and unnecessary: There's not any ambiguity in the original choice of symbols.