Is the theory of $R$-modules a first order theory?

Question

Is the theory of $R$-modules over a fixed ring $R$ a first order theory? (If so, can I use the upward Löwenheim–Skolem Theorem to show that there are arbitrary large $R$-modules of a given cardinality?)

Answer 1

The theory of $R$-modules can be seen as a first-order theory as follows: The language consists of the constant $0_M$, a unary operation $m_a$ for each $a \in R$, and a binary operation $+$, satisfying the following axioms:

1. $\forall x,y,\, x+y=y+x$
2. $\forall x,\, x+0_M=x$
3. $\forall x,y,z,\, x+(y+z)=(x+y)+z$.
4. $\forall x,\, \exists y,\, x+y=0_M$.
5. For each $a \in R$, $m_a0_M=0_M$.
6. For each $a,b \in R$, $\forall x,\,m_ax+m_bx=m_{a+b}x$.
7. For each $a,b \in R$, $\forall x,\, m_b(m_ax)=m_{ba}x$.
8. For each $a\in R$, $\forall x,y,\, m_a(x+y)=m_ax+m_ay$.
9. $\forall x,\, m_1x=x$.