Realizing a quotient ring as an abelian group Show that $\mathcal{M} = \mathbb{Z}[X] /(X^{2} - X, 4X +2)$ is finitely generated abelian group. 
What is the general procedure for handling these types?

Let $R=\Bbb Z[X]/(X^2-X)$.
Firstly $X$ is idempotent in the quotient and  $R=XR\oplus(1-X)R$. 
Since everything in the quotient can be reduced to a coset of the form $a+bX + (X^2-X)$ for integers a, b,  we have $XR=X\Bbb Z$ Mod $X^2-X$.
Since $(1-X)(a+bX)=a-aX=a(1-X)$, we have $(1-X)R=(1-X)\Bbb Z$. So,'you have that $R& is a direct sum of two cyclic Abelian groups, and that qualifies as a finitely generated Abelian group.
The ring you're talking about is a quotient of this, hence is a $f.g. Abelian group as well.
