V N U . JO U R N A L OF SC IE N C E , M a th e m a tic s - Physics.

T.xx,

N q2 - 2004

S M A L L M O D U L E S A N D Q F -R IN G S

N g o Si T u n g
Department o f Mathematics, Vinh University
A b s t r a c t . It is shown that a semiperfect ring R is quasi-Frobenius if and only if R has
finite right uniform dimension and every closed uniform submodule of R(u ) is a direct
summand where R(u>) denotes the direct sum of UJ copies of the right R -module R and
u is the first infinite ordinal. This result extends the one of D. V. Huynh and N. s. Tung
in [5. Theorem 1].
1. I n t r o d u c t i o n
Quasi-Frobenius rings (briefly, a QF-ring) were introduced by Nakayama in 1938.
A ring R is a Q F if it is a left artinian, left seflinjective ring. The class of QF-rings is one
of the most interesting generlization of semisimple rings and have been studied by several
authors (see, for example [4], [5], [7]). The number of characterization of QF-rmgs are so
large th at we are unable to give all the references here. In this paper, we will extend the
result which was given by D. V. H uynh and N. s. Tung in [5]. Throughtout this note all
rings R are associative rings w ith indentity and all modules are unitary right /ỉ-inođules.
2. P r e l i m i n a r i e s
A submodule N of a module M is called small in M , or a small submodule of M,
denoted by N c ° M , if for each submodule II of M, the relation N + II = M implies
II = M (or equivalently for each proper submodule II of M, M ^ N + II). A module s is
said to be a small module, if s is small in its injective hull. If s is not a small module, we
say th a t s is non-small. By this definition we may consider the zero module as a non-small
module although it is small in each non-zero module.
Small modules a n d non-small modules have been considered by many authors. In

u©

k e r 7Tjfc a s d e s i r e d .

Case 2: /Ỉ satisfies 6).
By the hypothesis ò), Ư is non-small, since eĩ R is a local module then each proper
submodule of CịR is small. Hence ư = e ị R 1 otherword eịR is uniform, we have i).
i) and iii) can prove similarly.
The following theorem was given by D. V. Huynh and N. s. Tung in [5].
T h e o r e m 4. Let R be a semiperfect ring. Then the following statements are equavelent:
i) R is a QF-ring.
ii) R has finite right uniform dimension, no non-zero projective right ideal of R is
contained in the jacobson radical J (R ) of R and every closed uniform submodule of R(u)
is a direct summand.
Now we prove our main theorem.
T h e o r e m 5. A seinipcrfect ring R is a QF i f and only i f R has finite right uniform
dimension and every closed uniform submodule of R(u>) is a direct summand.
Proof. (=>) Suppose that semiperfect ring R is a QF-ring. Then every closed submodule
of R ( uj) is non-small, by [6, Theorem 24.20]. By Lemma 3, each e iR is uniform, hence R
has finite right uniform dimension and each closed uniform submodule of R(uj) is a direct
summand.
(
the maximal submodule ( M ® T \ ) Ị T \ . Therefore
(T* © Ĩ \ ) / T i = ( P k Q T ^ / T u
implying T* ® 7\ = p k ® 7\ . Hence p k + T* = T* + T \ . Now by m odularity we have
B n (Pfc 4- T*) = ( B n p k) + T* = T + T*
= T* = B n ( r ® T ! )
= T* ® ( B n T i ) .
Consequently B n Tị = 0, a contradiction to the fact th a t T\ 7^ 0 and B IS esstial
in R{u>). Thus B = R{ uj), as desired.


Sm a ll m o dules a n d Q F - r in g s

43

By [6, Theorem 2.25], every local direct summand of R(w) is a direct summand.
We use this to show below th a t every closed submodule of R( uj) is a direct summand.
Let A be a non-zero submodule of R ( uj), with 0 / a e A. Then aR is a xyclic
submodule. Hence there exits a finite subset F of I such that a R c ®j £FR j .
From this it follows th a t a R has finite uniform dimension, so A contains a uniform
submodule.
Let Q be a non-zero closed submodule of R(u). Then Q contains a closed uniform
submodule Ư which is also closed in R (u ). Hence u is a direct summand of R ( uj), by the
hypothesis.

Let
JC = {A = (BkGKƯkị Uk is a uniform submodule of Q, A = (BkeKƯk
is a local direct summand}.
By the above argum ent /C Ỷ 0*
Prom this we may use Zorn’s Lemma to that /C contains a maximal element L =
®k€ỉ