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?