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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
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 “for a ring R , all left R -modules are serial if and only if R 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 R -module M is called prime uniserial ( ℘ -uniserial ) if its prime submodules are linearly ordered with respect to inclusion, and an R -module M is called prime serial ( ℘ -serial )if M is a direct sum of ℘ -uniserial modules. In this paper, it is shown that the ℘ -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 ℘ -serial module is serial. Let R be Morita equivalent to a commutative ring S . It is shown that every ℘ -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 ℘ -serial left R -module is serial if and only if R is serial ring with dim( R ) ≤ 1

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