Journal of Pure and Applied Algebra 212 (2008) 2092–2104
www.elsevier.com/locate/jpaa
Monogeny dimension relative to a fixed uniform module
Alberto Facchini
a,∗
, Pavel P
ˇ
r
´
ıhoda
b
a
Dipartimento di Matematica Pura e Applicata, Universit
`
a di Padova, 35121 Padova, Italy
b
Charles University in Prague, Faculty of Mathematics and Physics, Department of Algebra, Sokolovsk
´
a 83, 186 75 Prague, Czech Republic
Received 10 May 2007; received in revised form 22 November 2007
Available online 4 March 2008
Communicated by G. Rosolini
Abstract
For a module A and a uniform module U, we consider the invariant m-dim
U
(A) := sup{i ∈ N
0
| there exist morphisms
f : U
i
→ A and g : A → U
i
with g f a monomorphism}. This invariant turns out to have the following properties:
(1) m-dim
U
(A ⊕ B) = m-dim
U
(A) + m-dim
U
(B) for every A, B ∈ Mod-R; (2) if U and V are uniform and [U]
m
= [V ]
m
, then
m-dim
U
= m-dim
V
; and (3) if A, B ∈ Mod-R have finite Goldie dimension and [ A]
m
= [B]
m
, then m-dim
U
(A) = m-dim
U
(B)
for every uniform module U. In particular, when A has finite Goldie dimension and is a direct summand of a serial module, the
values m-dim
U
(A) completely determine the monogeny class of the module A. We give a complete description of the monoid of
all isomorphism classes of serial modules of finite Goldie dimension over a fixed ring R.
c
2008 Elsevier B.V. All rights reserved.
MSC: 16D70
1. Introduction
For an associative ring R with identity, we denote by Mod-R the class of all right modules over R. Two right
modules A and B over R are said to have the same monogeny class if there are a monomorphism A → B and a
monomorphism B → A. Similarly, A and B are said to have the same epigeny class if there exist an epimorphism
A → B and an epimorphism B → A. For any fixed R-module A, we shall denote by A, [ A]
m
and [ A]
e
the
isomorphism class, the monogeny class and the epigeny class of A, respectively, that is, A = {B ∈ Mod-R |
A
∼
=
B}, [A]
m
= {B ∈ Mod-R | there exist a monomorphism A → B and a monomorphism B → A}, and
[A]
e
= {B ∈ Mod-R | there exist an epimorphism A → B and an epimorphism B → A}. Recall that an R-module A
is uniform if it has Goldie dimension one, that is, if it is non-zero and the intersection of any two non-zero submodules
of A is non-zero. If U
1
, . . . , U
t
, V
1
, . . . , V
s
are uniform modules, then
[U
1
⊕ · · · ⊕ U
t
]
m
= [V
1
⊕ · · · ⊕ V
s
]
m
if and only if t = s and there is a permutation σ of {1, 2, . . . , t} such that [U
i
]
m
= [V
σ (i)
]
m
for every i = 1, 2, . . . , t
(see [1]). Thus if we denote by SUfm the class of all right R-modules that are direct sums of finitely many uniform
∗
Corresponding author.
E-mail address: facchini@math.unipd.it (A. Facchini).
0022-4049/$ - see front matter
c
2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.jpaa.2007.11.014