Rings all of whose prime serial modules are serial

Abstract

It is well known that the concept of left serial ring is a Morita invariant property and a theorem due to Nakayama and Skornyakov states that “for a ring [Formula: see text], all left [Formula: see text]-modules are serial if and only if [Formula: see text] is an Artinian serial ring”. Most recently the notions of “prime uniserial modules” and “prime serial modules” have been introduced and studied by Behboodi and Fazelpour in [Prime uniserial modules and rings, submitted; Noetherian rings whose modules are prime serial, Algebras and Represent. Theory 19(4) (2016) 11 pp]. An [Formula: see text]-module [Formula: see text] is called prime uniserial ( [Formula: see text]-uniserial) if its prime submodules are linearly ordered with respect to inclusion, and an [Formula: see text]-module [Formula: see text] is called prime serial ( [Formula: see text]-serial) if [Formula: see text] is a direct sum of [Formula: see text]-uniserial modules. In this paper, it is shown that the [Formula: see text]-serial property is a Morita invariant property. Also, we study what happens if, in the above Nakayama–Skornyakov Theorem, instead of considering rings for which all modules are serial, we consider rings for which every [Formula: see text]-serial module is serial. Let [Formula: see text] be Morita equivalent to a commutative ring [Formula: see text]. It is shown that every [Formula: see text]-uniserial left [Formula: see text]-module is uniserial if and only if [Formula: see text] is a zero-dimensional arithmetic ring with [Formula: see text] T-nilpotent. Moreover, if [Formula: see text] is Noetherian, then every [Formula: see text]-serial left [Formula: see text]-module is serial if and only if [Formula: see text] is serial ring with dim[Formula: see text].