Morley rank of group This is an example from S.Buechler's book Essential Stability Theory.
Let $M$ be the group $\bigoplus_{i<\omega}(\mathbb{Z}_4)_{i}$ with $\mathbb{Z}_4=\mathbb{Z}/4\mathbb{Z}$. Suppose $M^{*}$ is the monster model of $Th(M)$. My questions are the following:

*

*Why $2M^{*}$ is a vector space over $\mathbb{Z}_2$?


*Why the Morley rank of $M^{*}$ is 2?
Any hints or comments to my questions are welcomed. Thank you!

For the first question: $M$ is an abelian group of exponent $4$, and hence so is $M^*$. It follows that $2M^*$ is an abelian group of exponent $2$. An abelian group of prime exponent $p$ is a vector space over the $p$-element field. (More generally, an abelian group of exponent $n>0$ is a $\mathbf Z/n\mathbf Z$-module.)
Regarding your second question, I suppose you want to compute the Morley rank of $M$ as a pure abelian group. The subgroup $2M\leq M$ is infinite, so it has Morley rank at least $1$, and $[M:2M]$ is infinite, so $M$ has Morley rank at least $2$.
To obtain the upper bound, it is enough to show that $M/2M$ and $2M$ are both rank $1$. But that follows from the fact that (for each of them), the induced structure is that of a vector space over the two-element field, and as such, they are strongly minimal.
