I was trying to compute some examples dealing with length of modules and got stuck with this simple example:
Let $R=k[t]/(t^2)$ where $k$ is a field and $J=(x^3)$ be the ideal of the polynomial ring $R[x]$. I want to compute the length $l_{R}(R[x]/J)$ and $l_{R[x]}(R[x]/J)$.
Using the Macaulay2 code, I found the length $l_{R[x]}(R[x]/J) =6$. (I don't know how to use Macaulay2 to compute $l_{R}(R[x]/J)$ though).
This is where I got confused. Is this true that the composition series of $R[x]/J$ over $R[x]$ is $0\subseteq (x^2) \subseteq (x) \subseteq R[x]/J$? If so, $l_{R[x]}(R[x]/J)=3$. Can we have a larger composition series here? If not, is it still true that if $R$ has finite length and $J$ is a monomial ideal in the polynomial ring, the length of $R[x_1,\ldots ,x_n]/J$ over $R[x]$ is the number of monomials that not in $J$ (as the case when $R$ is a field)? And I can't see why we can have a length-6 composition series for $R[x]/J$ over $R$.
Thank you in advance for any help!
In your series ideal $(x)$ is not even maximal: $R[x]/(x)=R$ which is not a field. A composition series of length six is $$ 0 \subset (x^2t) \subset (xt) \subset (x^2) \subset (x^2, t) \subset (x, t) \subset R[x]/J. $$