A-module and left regular A-module

Question

My question is: what's the difference between an A-module and a left regular A-module $_A A$? My teacher gave me these definition but they seem me the same.

$A$-module $M$:

Let $A$ be a ring. A left $A$-module is an abelian (additive) group $M$ with the map $A\times M\rightarrow M$ defined by $(a,m)\rightarrow am$ called external product with the following properties:

(1) $(a_1+a_2)m=a_1m+a_2m$.
(2) $a(m_1+m_2)=am_1+am_2$.
(3) $1_am=m$.
(4) $(a_1 a_2)m=a_1(a_2m)$.

Left regular $A$-module $_A A$:

Let $A$ be a ring, $M=(A,+)$, a map $A\times M\rightarrow M$ with $(a,x)\rightarrow ax$. We obtain an $A$-module that we call left regular A-module $_A A$.

Thank you!

Answer 1

"The regular left $A$ module $_AA$" is just a particular $A$-module, where the set $M=A$ and the module operations are those of $A$.

There is, of course, a regular right $A$ module $A_A$, and that is a particular right $A$ module.