Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules

Abstract

Let $R$ be a commutative ring. A submodule $N$ of an $R$-module $M$ is semiprime if $m\in(N:m)M$ implies $m\in N$. It is known that every semiprime submodule of $R^n$ is an intersection of prime submodules, and Aryapoor extended this to finitely generated modules. We show that no finiteness assumption is needed: every semiprime submodule of an \emph{arbitrary} $R$-module is the intersection of the prime submodules containing it. The proof is short and uses neither coordinates nor presentations; its key step is that a nonzero element $q$ of a semiprime module $Q$ satisfies $sq\notin\p Q$ for all $s\notin\p$, where $\p$ is any minimal prime over the annihilator of $q$. We define an $R$-module $M$ to be real if $\sum_i m_i\otimes m_i=0$ in $M\otimes_RM$ implies $m_i=0$ for all $i$, and a submodule $N\subseteq M$ real if the module $M/N$ is real. We prove the corresponding result for real submodules: every real submodule of an arbitrary module is the intersection of the real prime submodules containing it. On $R^n$ our notion of realness coincides with the coordinate definition used in earlier work on real Nullstellensätze for free modules.