When defining noetherian ring/module there's no condition on the number of generators of ideals/submodules (apart from being finite).
However, in some cases we can do better:
-A noetherian module over a field is a finite vector space, so every submodule can be generated with at most n elements.
-A maximal ideal of $\mathbb{K}[X_1,…,X_n]$ where $\mathbb{K}$ is algebraically closed can be generated, via Nullstellensatz, by exactly n generators.
What other examples are there where we can find a system of generators of bounded cardinality?
What happens if we replace maximal ideal by prime ideal in the second example?
Best Answer
Understanding the number of generators is a very subtle problem. I will focus on your second question on ideals, since the first one is a bit broad. By a theorem of Foster-Swan, the problem is local.
There is no absolute upper bound even for a prime ideal of height $2$ in $k[[x,y,z]]$. In this paper, Moh gives a sequence of primes $(P_n)$ such that $\mu(P_n) =n+1$.
OK, so what to do next? One can ask if there are good bounds on $\mu(I)$ if $R/I$ is "nice". If $R/I$ is a complete intersection, then $\mu(I)$ is the height of $I$. In commutative algebra, the next level of "niceness" is being Gorenstein. In this paper Schoutens shows that if $I$ has height $2$ and $R/I$ Gorenstein, then there is a bound only depending on $R$.
If one goes down another notch, and only assume $R/I$ is Cohen-Macaulay, then Moh's examples show there are no hope for bounds independent of $I$. However, there are bounds that depends on invariants of $R/I$, such as the type or multiplicity.