Let $(A,\mathfrak{m})$ be a local Artinian ring with finite residue field, which I'm happy to assume is $\mathbf{F}_3$. (In particular, $A$ has finitely many elements.)

I would like to do some computations of the following kind, as $I$ ranges over all of the ideals of $A$.

(0) A way to enumerate all the ideals of $A$.

(1) For an ideal $I$ of $A$, compute the length of $I/I^2$.

(2) For an ideal $I$ of $A$, compute the ideal $J = \mathrm{Ann}(I)$.

(3) For an ideal $I$ of $A$, decide if $I$ is principal. (By computing the length of $I/\mathfrak{m} I$ or otherwise.)

The ring $A$ itself will be given explicitly as a quotient of a power series ring over $W(\mathbf{F}_3) = \mathbf{Z}_3$. For example, $A$ might be given as $\mathbf{Z}_3[[x]]/(27,9x,x^3)$ or $\mathbf{Z}_3[[x]]/(9,x^2)$.

My question: What is the computer algebra package that is best suited to carry out these computations? (I would like something that can be semi-automated for various possible $A$.) I would be interested in even a very simple one like $\mathbf{Z}_3[[x]]/(9,x^2)$

EDIT 2: There seems to be a consensus in the comments that this problem is significantly more manageable if $A$ is actually an algebra over its residue field. For example, in MAGMA, it is only possible to create ideals and quotient rings in univariate polynomial rings over fields. Other computer algebra packages have similar issues when the coefficient ring is not a field, although SINGULAR (for example) has some functionality with polynomials in several variables. As it happens, the problem I was interested in studying is still of interest for such fields.

b^2,b^3) I = ideal (a+2*b, b^2) ann I The output is (b^2, ab, a^2). $\endgroup$ – Sam Lichtenstein Aug 21 '10 at 21:19