What is an $R$-algebra?

Question

In the following, assume that rings are rings with unity.

Here is the definition of $R$-algebra from Wikipedia:

Let $R$ be a commutative ring and $(M,+,\cdot)$ an $R$-module.

Assume $\ast$ is a binary operation on $M$, such that:

1. $x\ast (y+z)= x\ast y + x\ast z$
2. $\forall x,y,z \in M.(x+y)\ast z = x\ast z + y\ast z$
3. $\forall r,s \in R,x,y \in M.(rx)\ast (sy)=(rs)(x\ast y) (r,s\in R, x,y\in M)$

then $(M,+,\cdot,\ast)$ is called an $R$-algebra.

This definition has the same form as my definition for "algebra over a field". Note that this definition does not require $\ast$ to be associative.

Here is an equivalent definition given in Dummit&Foote (original form is given using a ring homomorphism):

Let $(M,+,\cdot)$ be a ring and $R$ a commutive ring.

If $M$ is an $R$-module and the multiplication on $M$ is bilinear, then $M$ is called "an $R$-algebra".

Note that this definition requires $\ast$ to be associative.

What is the usual definition for $R$-algebra?

References

Abstract Algebra, Dummit and Foote, 3rd edition, page 355.