Do modules have to be defined over rings with unity?

Question

This is definition for left module over a ring $R$ given in Wikipedia:

Suppose that $R$ is a ring and $1_R$ is its multiplicative identity. A left $R$-module $M$ consists of an abelian group $(M, +)$ and an operation $\cdot\;: R \times M \to M$ such that for all $r, s \in R$ and $x, y \in M$, we have:

1. $r \cdot ( x + y ) = r \cdot x + r \cdot y$

2. $( r + s ) \cdot x = r \cdot x + s \cdot x $

3. $ ( r s ) \cdot x = r \cdot ( s \cdot x )$

4. $1_R \cdot x = x $

But then it necessary that the ring $R$ should be a ring with unity. Then why it is not mentioned in definition ?

Answer 1

Some authors use the term "ring" to mean "ring with identity." This is neither standard nor nonstandard; there is just no consensus. Authors using this definition would use the term "rng" to denote a ring that possibly does not have an identity.

Answer 2

When that definition says

Let $R$ be a ring and $1_R$ its multiplicative identity...

they are mentioning (implicitly) that they are requiring $R$ to be unital.

However, it is certainly possible to define modules over a non-unital ring; just throw out statement 4 from that definition.