Composition Series of $R$-modules Recently I have been reading about composition series and lengths of modules. Everything that I have encountered has been rather sparse on examples. In this vein, I was wondering how you would compute the following: 
Let $R=\mathbb{Z}[T]$ be the ring of polynomials in one variable.
(i) Composition series and length of the $R$-module $R/(T^{2}-T+1,3)$
(ii) Composition series and length of the $R$-module $R/(T^{2}-T+1,28)$
Any help would be appreciated!
 A: a) We have $(T^2-T+1,3)=((T+1)^2,3)$. This shows that $\mathfrak m=(T+1,3)$ is the only prime (maximal) ideal of $R$ containing $(T^2-T+1,3)$. You are in the following situation: the $R$-module is $M=R/I$ with $\mathfrak m^2\subset I\subset\mathfrak m$. Then a composition series is $0<\mathfrak m/I<R/I$. 
(Actually, if you have a local artinian ring $S$ whose maximal ideal $\mathfrak n$ satisfies $\mathfrak n^2=0$, then $l(S)=1+\dim_{S/\mathfrak n}\mathfrak n$, and $\dim_{S/\mathfrak n}\mathfrak n$ is nothing but the minimal number of generators of $\mathfrak n$. This remark can be easily generalized to the case $\mathfrak n^r=0$, $r>2$.)
b) If I'm not mistaken in this case there are three maximal ideals containing $(T^2-T+1,28)$, namely $\mathfrak m_1=(T^2-T+1,2)$, $\mathfrak m_2=(T-3,7)$ and $\mathfrak m_3=(T+2,7)$. Furthermore, $\mathfrak m_1^2\mathfrak m_2\mathfrak m_3=0$ (in $R/I$) and starting from this we get $$l(R/I)=1+l(\mathfrak m_1/\mathfrak m_1^2)+l(\mathfrak m_1^2/\mathfrak m_1^2\mathfrak m_2)+l(\mathfrak m_1^2\mathfrak m_2)$$ and all these are vector spaces, so their length equals the dimension.
