Minimal Prime and Semiprime Submodules

Prime and semiprime submodules are important generalizations of prime and semiprime ideals to module theory over commutative rings. However, minimal or smallest prime/semiprime submodules have received comparatively less attention. This paper furthers the theory of minimal prime and minimal semiprime submodules through several new directions. After formally introducing these concepts, we establish characterization theorems for minimal prime submodules based on intersections of primes and associated primes. A complete structure theory is also developed for minimal semiprime submodules, allowing their description up to isomorphism. We introduce prime and semiprime operators which stably preserve primeness properties between modules. Results on the persistence of minimal primeness of submodules under such operators are proved. Connections of our minimal semiprime characterization back to classic ring theory are highlighted, recovering past results on minimal prime ideals. Overall, this paper significantly expands the foundations of minimal prime and semiprime submodules, while opening up many new questions at this nexus of module and ring theory.