Any ring \(R\) with Krull dimension (i.e., \(k(R)\) exists) satisfies the ascending chain condition (ACC) on semiprime ideals [R. Gordon and J. C. Robson, Krull dimension, Mem. Am. Math. Soc. 133 (1973; Zbl 0269.16017), Theorem 7.6]. This result does not hold more generally for modules.
In this paper, the authors give the following result: If \(R\) is the first Weyl algebra over a field of characteristic \(0\), then there are Artinian \(R\)-modules which do not satisfy the ascending chain condition on prime submodules.
However, the authors prove the following important result: If \(R\) is a ring which satisfies a polynomial identity, then any \(R\)-module with Krull dimension satisfies the ascending chain on prime submodules, and, if \(R\) is left Noetherian, also the ascending chain condition on semiprime submodules.