About the definition of a module.

Question

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?

Answer 1

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.

Answer 2

It's true that the $+$ in $r + s$ is addition in $R$, while the $+$ in $x + y$ is addition in $M$ (while the $+$ in both $rx + ry$ and $rx + sx$ is addition in $M$), and to be perfectly precise, one should use different symbols, perhaps $+_R$ and $+_M$. On the other hand, using the same symbol in an unambiguous context is harmless: you can think of this as an instance of "operator overloading" if you know this concept from programming languages. Which binary operation is meant can always be inferred from the structure containing the operands.

Note that this notational compromise would not be possible if $R$ were a subset of $M$ but not a subgroup; in this case, it would be necessary to use entirely different symbols so as not to confuse the two possible operations on $r + s$, considered either as a sum of elements of $R$ or of $M$. Fortunately this situation is quite uncommon, and just like we usually omit the multiplication symbol in a product such as $rs$, we usually use $+$ for all abelian group operations in modules.