## “School of Mathematics”

Back to Papers Home
Back to Papers of School of Mathematics

Paper   IPM / M / 14593
 School of Mathematics Title: Rings all of whose prime serial modules are serial Author(s): M. Behboodi (Joint with Z. Fazelpour, and M. R. Vedadi) Status: Published Journal: J. Algebra Appl. No.: 2 Vol.: 16 Year: 2017 Pages: 1-20 Supported by: IPM
Abstract:
t 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 âfor a ring R , all left R -modules are serial if and only if R is an Artinian serial ringâ. Most recently the notions of âprime uniserial modulesâ and âprime serial modulesâ 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 R -module M is called prime uniserial ( â -uniserial ) if its prime submodules are linearly ordered with respect to inclusion, and an R -module M is called prime serial ( â -serial )if M is a direct sum of â -uniserial modules. In this paper, it is shown that the â -serial property is a Morita invariant property. Also, we study what happens if, in the above NakayamaâSkornyakov Theorem, instead of considering rings for which all modules are serial, we consider rings for which every â -serial module is serial. Let R be Morita equivalent to a commutative ring S . It is shown that every â -uniserial left R -module is uniserial if and only if R is a zero-dimensional arithmetic ring with J ( R ) T-nilpotent. Moreover, if S is Noetherian, then every â -serial left R -module is serial if and only if R is serial ring with dim( R ) â¤ 1