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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 âfor a ring
R
, all left
R
-modules
are serial if and only if
R
is an Artinian serial ringâ. Most recently the notions of âprime
uniserial modulesâ and âprime 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 NakayamaâSkornyakov 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
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