When is a Nakayama closure semiprime?

Abstract

Many well-known closure operations such as integral closure and tight closure are both semiprime operations and Nakayama closures. In this short note, we begin the study on the overlap between Nakayama closures and semiprime operations. We exhibit examples of closure operations which are either semiprime or Nakayama but not the other. In the case of a discrete valuation ring we show that a closure operation $c$ is Nakayama if and only if it is semiprime and \[ (0)^c=\bigcap_{n \geq 1} (I^n)^c \] for any ideal $I$.