A generalization of semiperfect modules
A module $M$ is called radical semiperfect, if $\frac MN$ has a projective cover whenever $\mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M$. We study various properties of these modules. It is proved that every left $R$-module is radical semiperfect if and only if $R$ is left perfect. Mo...
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| author | Türkmen, B. N. Тюркмен, Б. Н. |
| author_facet | Türkmen, B. N. Тюркмен, Б. Н. |
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| description | A module $M$ is called radical semiperfect, if $\frac MN$ has a projective cover whenever $\mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M$. We study various properties of these modules. It is proved that every left $R$-module is radical semiperfect if and only if $R$ is left perfect.
Moreover, radical lifting modules are defined as a generalization of lifting modules. |
| first_indexed | 2026-03-24T02:10:29Z |
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UDC 512.5
B. N. Türkmen (Amasya Univ., Turkey)
A GENERALIZATION OF SEMIPERFECT MODULES
УЗАГАЛЬНЕННЯ НАПIВДОСКОНАЛИХ МОДУЛIВ
A module M is called radical semiperfect, if
M
N
has a projective cover whenever \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M . We study various
properties of these modules. It is proved that every left R-module is radical semiperfect if and only if R is left perfect.
Moreover, radical lifting modules are defined as a generalization of lifting modules.
Модуль M називається радикальним напiвдосконалим, якщо
M
N
має проективне покриття, як тiльки \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq
\subseteq N \subseteq M . Дослiджено рiзнi властивостi цих модулiв. Доведено, що кожен лiвий R-модуль є радикально
напiвдосконалим тодi i тiльки тодi, коли R є лiвим досконалим. Крiм того, радикальнi пiднiмаючi модулi визначено,
як узагальнення пiднiмаючих модулiв.
1. Introduction. Throughout this paper, all rings are associative with identity element and all
modules are unital left R-modules. Let M be a left module over such a ring R. The notation
(N \subset M ) N \subseteq M means that N is a (proper) submodule of M. A submodule N \subset M is said to
be small in M, denoted as N \ll M, if M \not = N + L for every proper submodule L of M. If every
proper submodule N of M is small, then M is called hollow. A finitely generated hollow module is
local.
For modules M and P, let f : P \rightarrow M be an epimorphism. f is called cover if \mathrm{k}\mathrm{e}\mathrm{r}(f) is small
in P. A projective module P together with a cover f : P \rightarrow M is called a projective cover of M.
By [1], rings whose (finitely generated) modules have a projective cover is (semi) perfect.
In [6], Kasch and Mares transferred the notions of (semi)perfect rings to modules. Let M be
an R-module. M is called semiperfect if every factor module of M has a projective cover. They
gave a characterization of semiperfect modules via supplemented modules. A module M is called
supplemented if every submodule N of M has a supplement, that is a submodule K minimal with
respect to M = N +K. A submodule K \subseteq M is supplement of N in M if and only if M = N +K
and N \cap K \ll K. If M is a semiperfect module, then it is supplemented. Also, the converse of this
fact is true in case the module is projective.
Zöschinger completely determined in [14] (Lemma 2.1) the structure of a module M with small
radical over local Dedekind domains. Here the radical \mathrm{R}\mathrm{a}\mathrm{d}(M) of a module M is the intersection
of all maximal submodules of the module M, or equivalently the sum of all small submodules of
M. Then, he considered the modules whose radical have a supplement. He called these modules
as a radical supplemented module. Upon this Büyükaşık and Türkmen call a module M strongly
radical suplemented (or briefly a srs-module) if every submodule containing radical has a supplement
in M [2].
As motivated by the above definitions, it is natural to introduce radical semiperfect modules. We
say that a module M is radical semiperfect if, whenever \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M,
M
N
has a projective
cover.
In this article, we provide some properties of these modules. We prove that every (finitely
generated) left R-module is radical semiperfect if and only if R is (semi)perfect. Radical semiperfect
c\bigcirc B. N. TÜRKMEN, 2017
104 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
A GENERALIZATION OF SEMIPERFECT MODULES 105
modules are srs, and a projective module M is radical semiperfect if and only if it is srs\oplus , i.e.,
for every submodule \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M, there exists a direct summand K of M such that K is
a supplement of N in M (see [8]). We show that every direct summand of a radical semiperfect
module is radical semiperfect. We also study radical lifting modules as a generalization of lifting
modules.
2. Radical semiperfect modules. It is well known that, a ring R is left perfect if and only if
every left R-module is semiperfect. We generalize this fact via radical semiperfect modules.
It is clear that every semiperfect module is radical semiperfect. But, in general, the converse is
false as the following example shows.
Recall that a module M is radical if M has no maximal submodules, that is, M = \mathrm{R}\mathrm{a}\mathrm{d}(M).
Lemma 2.1. Every radical module is radical semiperfect.
Proof. Let M be a radical module. Then \mathrm{R}\mathrm{a}\mathrm{d}(M) = M. Therefore, the factor module
M
M
of M is zero. It follows that M is radical semiperfect.
By P (M) we will denote the sum of all radical submodules of a module M. Note that P (M) is
the largest radical submodule of M and P (M) \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(M). Using Lemma 2.1 we obtain that P (M)
is radical semiperfect for every left R-module M.
Example 2.1. Consider the left \BbbZ -module M =\BbbZ \BbbQ . Since \mathrm{R}\mathrm{a}\mathrm{d}(M) =M, we obtain that M is
a radical semiperfect module by Lemma 2.1. On the other hand, M is not semiperfect because it is
not supplemented [12].
Proposition 2.1. Every radical semiperfect module with small radical is semiperfect.
Proof. Let M be a radical semiperfect module with \mathrm{R}\mathrm{a}\mathrm{d}(M) \ll M and U be any submodule of
M. Then \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq U +\mathrm{R}\mathrm{a}\mathrm{d}(M). Since M is radical semiperfect,
M
U +\mathrm{R}\mathrm{a}\mathrm{d}(M)
has a projective
cover. By [11] (Proposition 2.1) there exists a submodule V of M such that M = (U+\mathrm{R}\mathrm{a}\mathrm{d}(M))+V,
(U + \mathrm{R}\mathrm{a}\mathrm{d}(M)) \cap V \ll V and V has a projective cover. Therefore, U \cap V is a small submodule
of V as the small submodule (U + \mathrm{R}\mathrm{a}\mathrm{d}(M)) \cap V in V. It follows from assumption that M =
= (U+\mathrm{R}\mathrm{a}\mathrm{d}(M))+V = U+V. Again by [11] (Proposition 2.1), we deduce that
M
U
has a projective
cover. Thus M is semiperfect.
Because of the above result, defining radical semiperfect rings do not make sense. That is, a ring
R is semiperfect if and only if R is radical semiperfect.
Proposition 2.2. Every radical semiperfect module is srs.
Proof. Let M be a radical semiperfect module and N be any submodule of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq
\subseteq N. Then the factor module
M
N
of M has a projective cover. Applying [11] (Proposition 2.1), N
has a supplement in M. Hence M is srs.
Proposition 2.3. Every factor module of a radical semiperfect module is radical semiperfect.
Proof. Let M be a radical semiperfect module and X \subseteq M. For any submodule
N
X
of
M
X
with
\mathrm{R}\mathrm{a}\mathrm{d}
\biggl(
M
X
\biggr)
\subseteq N
X
, we can write \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N because
\mathrm{R}\mathrm{a}\mathrm{d}(M) +X
X
\subseteq \mathrm{R}\mathrm{a}\mathrm{d}
\biggl(
M
X
\biggr)
\subseteq N
X
. Since
M is a radical semiperfect,
M
N
has a projective cover. Note that
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
106 B. N. TÜRKMEN
M
N
\sim =
M
X
N
X
and so
M
X
N
X
has a projective cover. Thus,
M
X
is radical semiperfect.
The following follows from Proposition 2.3.
Corollary 2.1. Every homomorphic image of a radical semiperfect module is radical semiperfect.
In particular, every direct summand of the module is radical semiperfect.
It is known in [10] (42.3 (2)-(ii)) that if K is semiperfect and \pi : M \rightarrow K is a cover, then M
is also semiperfect. Now, we give an analogous characterization of this fact for radical semiperfect
modules.
Lemma 2.2. Let K be a radical semiperfect module and let \pi : M \rightarrow K be a cover of K. Then
M is a radical semiperfect module.
Proof. Let N be a submodule of M such that \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. We have an epimorphism \varphi :
M
N
\rightarrow N
\pi (N)
defined by m + N \rightarrow \pi (m) + \pi (N). Since \pi : M \rightarrow K is a cover, we have
\mathrm{k}\mathrm{e}\mathrm{r}(\varphi ) \ll M
N
. Moreover, \mathrm{R}\mathrm{a}\mathrm{d}(K) \subseteq \pi (N). Since K is radical semiperfect, the factor module
K
\pi (N)
has a projective cover, say \psi : P \rightarrow K
\pi (N)
. Since P is projective, there exists a homomorphism h :
P \rightarrow M
N
such that \varphi \circ h = \psi . So h is an epimorphism. It follows from [10] (19.3 (1)) that h is a
cover. Therefore, projective module P together with a cover h is a projective cover of
M
N
. It means
that M is radical semiperfect.
We have the following corollary by Proposition 2.3 and Lemma 2.2. Note that a submodule
L \subseteq M is small in M if and only if the canonical homomorphism \pi : M - \rightarrow L is a cover of
M
L
.
Corollary 2.2. Let M be an R-module and L\ll M. Then M is radical semiperfect if and only
if
M
L
is radical semiperfect.
Theorem 2.1. Let M be a projective R-module. Then M is srs\oplus if and only if it is radical
semiperfect.
Proof. (=\Rightarrow ) Let N be any submodule of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. By the assumption, we can
write M = N +K, N \cap K \ll K and M = K \oplus K \prime for some submodules K,K \prime \subseteq M. Since M is
projective, we get that K is projective as a direct summand of M. Then a homomorphism f = \pi \circ i :
K \rightarrow M
N
is an epimorphism for the inclusion homomorphism i : K \rightarrow M and the canonical
homomorphism \pi : M \rightarrow M
N
. Since \mathrm{K}\mathrm{e}\mathrm{r}(\pi ) \ll K, K together with a cover f is a projective cover
of
M
N
. Hence, M is radical semiperfect.
(\Leftarrow =) Let N be any submodule of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. Since M is radical semiperfect,
M
N
has a projective cover. Let P together with a cover \varphi be a projective cover of
M
N
. Since M is
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
A GENERALIZATION OF SEMIPERFECT MODULES 107
projective, there exists a homomorphism f : M \rightarrow P such that \varphi \circ f = \pi , where \pi : M \rightarrow M
N
is
the canonical homomorphism. Since \varphi is a cover, f is an epimorphism by [10] (19.2). Since P
is projective, the exact sequence 0 \rightarrow N \rightarrow M \rightarrow P \rightarrow 0 is decomposable. Then there exists a
homomorphism g : P \rightarrow M such that f \circ g = 1P . Therefore, M = \mathrm{k}\mathrm{e}\mathrm{r} f \oplus g(P ) and \mathrm{k}\mathrm{e}\mathrm{r} f \subseteq N.
It follows that M = N + g(P ). On the other hand, \varphi = \pi | g(P ) : g(P ) \rightarrow M
N
is a cover. Thus
\mathrm{k}\mathrm{e}\mathrm{r}(\pi | g(P )) = U \cap g(P ) \ll g(P ). It means that M is a srs\oplus -module.
Corollary 2.3. Let M be an R-module and let P together with a cover f : P \rightarrow M be a
projective cover of M. Then the following statements are equivalent:
(1) M is a radical semiperfect module,
(2) P is a radical semiperfect module,
(3) P is a srs\oplus -module.
Proof. (1) \Leftarrow \Rightarrow (2). The necessity is clear from Lemma 2.2. Conversely is clear from
Corollary 2.1.
(2) \Leftarrow \Rightarrow (3). It follows from Theorem 2.1.
Proposition 2.4. Let M be a radical semiperfect module. Assume that M has a projective
cover. Then M is semiperfect.
Proof. Let f : P \rightarrow M be a projective cover. By Corollary 2.3, we obtain that P is a srs\oplus -
module. Applying [8] (Proposition 2.2) and [10] (42.1), P is semiperfect and then M is semiperfect
as a factor module of the module P according to [10] (42.3 (2)-(i)).
Recall from [4] that an R-module M is cofinitely semiperfect if every finitely generated factor
module of M has a projective cover. Now we give some relations between radical semiperfect
modules and cofinitely semiperfect modules.
Example 2.2 (see [5], (11.3)). Let R be the ring K[[x]] of all power series
\sum \infty
i=0
kix
i in an
indeterminate x and with coefficients from a field K. Then R is semiperfect but not left perfect. So
there exists an infinite index set I such that the left R-module R(I) is not a srs\oplus -module. Since
R(I) is projective, R(I) is not radical semiperfect by Theorem 2.1. By [4] (Theorem 2.12), R(I) is
cofinitely semiperfect.
Example 2.3. Let R be a local Dedekind domain with quotient field K. Then
M
L
\sim = K =
= \mathrm{R}\mathrm{a}\mathrm{d}(K) as an R-module, for a free R-module M and L \subseteq M. Since R is hereditary, M
is projective. It is clear that M is not \oplus -supplemented. So M is not a srs\oplus -module by [8]
(Proposition 2.9). By Theorem 2.1, M is not radical semiperfect. In addition, since R is semiperfect,
M is cofinitely semiperfect by [4] (Theorem 2.12).
In the following proposition, we prove that every cofinitely semiperfect module is radical semiper-
fect under a condition.
Proposition 2.5. Let M be a cofinitely semiperfect module. If
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
is finitely generated,
then M is radical semiperfect.
Proof. Let N be any submodule of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. Then
M
N
is finitely generated as a
factor module of the finitely generated module
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
. Since M is cofinitely semiperfect,
M
N
has
a projective cover. Thus, M is radical semiperfect.
Now, we give a characterization of perfect rings via radical semiperfect modules.
Theorem 2.2. A ring R is left perfect if and only if every left R-module is radical semiperfect.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
108 B. N. TÜRKMEN
Proof. (=\Rightarrow ) It is clear.
(\Leftarrow =) By Proposition 2.2 and [2] (Corollary 2.6).
Let R be an arbitrary ring. R is called a left Bass ring if every non zero left R-module has a
maximal submodule [3] (2.19).
Proposition 2.6. Let R be a ring. Then R is a left Bass ring if and only if every radical
semiperfect left R-module is semiperfect.
Proof. (=\Rightarrow ) Let M be any radical semiperfect left R-module. Since R is a left Bass ring, it
follows from [3] (2.21) that \mathrm{R}\mathrm{a}\mathrm{d}(M) \ll M. By Proposition 2.1, M is semiperfect.
(\Leftarrow =) Let M be a left R-module which is radical. By Lemma 2.1, M is radical semiperfect.
It follows from the assumption that it is semiperfect. If M \not = 0, then M has a maximal submodule
according to [11] (Lemma 3.7), a contradiction. So M = 0. It means that R is a left Bass ring.
Recall from [11] that a module M is called quasi semiperfect if every factor module of M has a
M -projective cover. As a proper generalization of this definition, we call a module M quasi radical
semiperfect if
M
N
has a M -projective cover for every submodule N of M containing \mathrm{R}\mathrm{a}\mathrm{d}(M).
It is clear that every radical semiperfect module is quasi radical semiperfect. But in general the
converse is not true. For example, \BbbZ \BbbQ is quasi radical semiperfect but not quasi semiperfect (see
Lemma 2.1 and [11], Theorem 2.4).
Lemma 2.3. Let M be a quasi radical semiperfect module. Then every submodule N of M
containing \mathrm{R}\mathrm{a}\mathrm{d}(M) has a supplement which has a M -projective cover.
Proof. Let N be any submodule of M containing \mathrm{R}\mathrm{a}\mathrm{d}(M). By the hypothesis,
M
N
has a
M -projective cover. By [11] (Proposition 2.3), N has a supplement which has a M -projective cover.
Now, we shall prove that a quasi radical semiperfect module is radical semiperfect under a certain
condition.
Theorem 2.3. Let M be a module with \mathrm{R}\mathrm{a}\mathrm{d}(M) \ll M. M is quasi radical semiperfect if and
only if it is quasi semiperfect.
Proof. (=\Rightarrow ) Let N be any submodule of M. Then \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N + \mathrm{R}\mathrm{a}\mathrm{d}(M). By the
hypothesis and Lemma 2.3, there exists a submodule V of M such that M = (N + \mathrm{R}\mathrm{a}\mathrm{d}(M)) + V
and (N + \mathrm{R}\mathrm{a}\mathrm{d}(M)) \cap V \ll V. Moreover, V has a M -projective cover. So we have M =
= (N + \mathrm{R}\mathrm{a}\mathrm{d}(M)) + V, (N + \mathrm{R}\mathrm{a}\mathrm{d}(M)) \cap V \ll V and V has a M -projective cover. Since
\mathrm{R}\mathrm{a}\mathrm{d}(M) \ll M, then M = N + V and N \cap V \ll V. By [11] (Proposition 2.3),
M
N
has a M -
projective cover. Thus M is quasi semiperfect.
(\Leftarrow =) It is clear from the definition.
Proposition 2.7. Every factor module of a quasi radical semiperfect module is quasi radical
semiperfect.
Proof. Let M be a quasi radical semiperfect module, X be any submodule of M and
N
X
be any
submodule of
M
X
such that \mathrm{R}\mathrm{a}\mathrm{d}
\biggl(
M
X
\biggr)
\subseteq N
X
. Then \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. By the hypothesis,
M
N
has a
M -projective cover. Therefore there exists a cover \varphi : P \rightarrow M
N
such that P is M -projective. Let \pi :
M \rightarrow M
X
be the canonical epimorphism. We have an epimorphism f :
M
X
\rightarrow M
N
. Then f \circ \pi :
M \rightarrow M
N
is an epimorphism. Since P is M -projective, there exists a homomorphism h : P \rightarrow M
such that f \circ \pi \circ h = \varphi . Then there exists a homomorphism \pi \circ h : P \rightarrow M
X such that f \circ \pi \circ h = \varphi .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
A GENERALIZATION OF SEMIPERFECT MODULES 109
Therefore, P is
M
X
-projective. Since
M
N
has a
M
X
-projective cover for every submodule
N
X
of
M
X
such that \mathrm{R}\mathrm{a}\mathrm{d}
\biggl(
M
X
\biggr)
\subseteq N
X
, then
M
X
is a quasi radical semiperfect module.
3. Radical lifting modules. Recall from [3] (22.2) that a module M is lifting if, for any
submodule N of M, there exists the decomposition M = N \prime \oplus K such that N \prime \subseteq N and N\cap K \ll K.
Every \pi -projective supplemented module is lifting. These modules are extensively studied by many
authors until now. For the properties and characterizations of lifting modules, we refer to [3].
In this section, we introduce the concept of radical lifting modules as a generalization of lifting
modules. We obtain the basic properties of such modules.
Definition 3.1. Let M be a module. A module M is called radical lifting if , for every submodule
N of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N, there exists the decomposition M = N \prime \oplus K such that N \prime \subseteq N and
N \cap K \ll K.
It is easy to see that as a equivalent condition of this definition; a module M is radical lifting if
and only if, for every submodule N of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N, N contains a direct summand K of
M such that
N
K
\ll M
K
.
Lemma 3.1. Let M be a radical lifting module. Then it is a srs\oplus -module.
Proof. Let N be any submodule of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. Since M is radical lifting, there are
a direct summand N \prime of M and a submodule K \leq M with N \prime \leq N, M = N \prime \oplus K and N\cap K \ll K.
Therefore M is a srs\oplus -module.
Let M be a nonzero module. M is called indecomposable if the only direct summands of M are
0 and M.
Corollary 3.1. Let M be an indecomposable module which is not radical. Then the following
statements are equivalent:
(1) M is radical lifting,
(2) M is lifting,
(3) M is local.
Proof. (3) =\Rightarrow (2) and (2) =\Rightarrow (1) are clear.
(1) =\Rightarrow (3). It follows from Lemma 3.1 and [8] (Lemma 2.3).
Lemma 3.2. Let M be a radical module. Then M is a radical lifting module.
Proof. Let N be any submodule of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. Since \mathrm{R}\mathrm{a}\mathrm{d}(M) =M, then N =M.
So we have M =M \oplus 0, M \leq N and N \cap 0 = 0 \ll 0. Thus, M is a radical lifting module.
The next result is a direct consequence of Lemma 3.2.
Corollary 3.2. P (M) is a radical lifting module for every R-module M.
The following fact is crucial.
Proposition 3.1. Let M = X\oplus L, where X is a radical module and L is a lifting module. Then
M is radical lifting.
Proof. For any submodule N of M with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N, we have X \subseteq N because X is radical.
Therefore, M = X\oplus L = N+L. Since L is lifting, there exists the decomposition L = L\prime \oplus V such
that L\prime \subseteq N \cap L and (N \cap L) \cap V = N \cap V \ll V. It follows that M = X \oplus L = X \oplus (L\prime \oplus V ) =
= (X \oplus L\prime )\oplus V = N +V. Note that X \oplus L\prime \subseteq N. Clearly, V is a supplement of N in M. It means
that M is radical lifting.
Every lifting module is radical lifting but the converse is not always true in the following example
shows.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
110 B. N. TÜRKMEN
Example 3.1. Let R be a local Dedekind domain with the with quotient field K. Put M = K(I)\oplus
\oplus
\biggl(
K
R
\biggr) (J)
\oplus R, where I and J are any infinite index sets. Therefore, P (M) = K(I) \oplus
\biggl(
K
R
\biggr) (J)
.
Since local modules are lifting, R is lifting and so M is radical lifting by Proposition 3.1. However,
it is not lifting according to [13] (Theorem 4.3).
In [7], a module M is said to be \oplus -supplemented if every submodule of M has a supplement
that is a direct summand of M. Under given definitions, we have the following diagram on modules:
lifting
xx &&
radical lifting
%%
\oplus -supplemented
xx ((
srs\oplus
''
supplemented
vv
srs
Corollary 3.3. Finitely generated radical lifting modules are \oplus -supplemented.
Proof. Let M be a finitely generated radical lifting module. Then, it is a srs\oplus -module by
Lemma 3.1. Since \mathrm{R}\mathrm{a}\mathrm{d}(M) is a small submodule of M, it follows from [8] (Lemma 2.2) that M is
\oplus -supplemented.
Theorem 3.1. Let M be a projective module. Then M is radical lifting if and only if it is lifting.
Proof. (=\Rightarrow ) Let M be a radical lifting module. By Lemma 3.1, M is srs\oplus . So M is
\oplus -supplemented by [8] (Lemma 2.2). Hence, M is lifting.
(\Leftarrow =) It is obvious.
Recall from [3] (27.8) that a supplemented module M is called strongly discrete if it is self-
projective.
Corollary 3.4. Let M be a projective module. Then the following statements are equivalent:
(1) M is strongly discrete,
(2) M is lifting,
(3) M is radical lifting,
(4) M is radical semiperfect.
Proof. (1) =\Rightarrow (2) and (2) =\Rightarrow (3) are clear.
(3) =\Rightarrow (4). By Lemma 3.1 and Theorem 2.1.
(4) =\Rightarrow (1). It follows from Theorem 2.1 and [8] (Lemma 2.2).
A module M is said to have the Summand Sum Property (SSP ) if the sum of two direct
summands of M is again a direct summand of M.
Proposition 3.2. Let M be a radical lifting module with the property (SSP ). Suppose that X
is a direct summand of M. Then
M
X
is a radical lifting module.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
A GENERALIZATION OF SEMIPERFECT MODULES 111
Proof. Let
N
X
be any submodule of
M
X
with \mathrm{R}\mathrm{a}\mathrm{d}
\biggl(
M
X
\biggr)
\subseteq N
X
. Since
\mathrm{R}\mathrm{a}\mathrm{d}(M) +X
X
\subseteq
\subseteq \mathrm{R}\mathrm{a}\mathrm{d}
\biggl(
M
X
\biggr)
, then \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. By the assumption, there exist submodules N \prime , K of M such
that M = N \prime \oplus K, N \prime \leq N and N \cap K \ll K. Thus,
K +X
X
is a supplement of
N
X
in
M
X
by [10] (41.1 (7)). Since M has the property (SSP ), there exists a submodule Y of M such that
M = (N \prime +X)\oplus Y. So we can write
M
X
=
\biggl(
N \prime +X
X
\biggr)
\oplus
\biggl(
Y +X
X
\biggr)
. It is clear that
N \prime +X
X
\leq N
X
.
Furthermore,
N
X
\cap Y +X
X
\ll Y +X
X
. Thus, the factor module
M
X
is radical lifting.
Recall from [10] (6.4) that a submodule X of an R-module M is called fully invariant if f(X)
is contained in X for every R-endomorphism f of M. Let M be an R-module and \tau be a preradical
for the category of R-modules. Then, \tau (M) is a fully invariant submodule of M. M is called a duo
module if every submodule of M is fully invariant [9].
Proposition 3.3. Let M be a radical lifting module. If X is a fully invariant submodule of M,
then
M
X
is a radical lifting module.
Proof. Let
N
X
be any submodule of
M
X
with \mathrm{R}\mathrm{a}\mathrm{d}
\biggl(
M
X
\biggr)
\subseteq N
X
. Since
\mathrm{R}\mathrm{a}\mathrm{d}(M) +X
X
\subseteq
\subseteq \mathrm{R}\mathrm{a}\mathrm{d}
\biggl(
M
X
\biggr)
, then \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N. By the hypothesis, there exists a direct summand N \prime of M
such that N \prime \leq N, M = N \prime \oplus K for a submodule K \leq M, and N \cap K \ll K. So we have
M
X
=
\biggl(
N \prime +X
X
\biggr)
\oplus
\biggl(
K +X
X
\biggr)
,
N \prime +X
X
\leq N
X
and
N
X
\cap K +X
X
\ll K +X
X
. Therefore the factor
module
M
X
is radical lifting.
Corollary 3.5. The following statements are equivalent for a duo module M :
(1) M is radical lifting,
(2) every direct summand of M is radical lifting.
Let M be a module. M is called reduced if P (M) = 0. Note that
M
P (M)
is reduced for every
left R-module M.
Lemma 3.3. Reduced radical lifting modules have a small radical.
Proof. Let M be a reduced radical lifting module. Then, there exist submodule X and N of
M such that M = X \oplus N, \mathrm{R}\mathrm{a}\mathrm{d}(M) \cap N = \mathrm{R}\mathrm{a}\mathrm{d}(N) \ll N and X \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(M). It follows that
\mathrm{R}\mathrm{a}\mathrm{d}(M) = \mathrm{R}\mathrm{a}\mathrm{d}(X) \oplus \mathrm{R}\mathrm{a}\mathrm{d}(N) = X \oplus \mathrm{R}\mathrm{a}\mathrm{d}(N). So X = \mathrm{R}\mathrm{a}\mathrm{d}(X), that is, X is radical. By the
assumption, we get X \subseteq P (M) = 0. It means that \mathrm{R}\mathrm{a}\mathrm{d}(M) \ll M.
Recall that a module M is coatomic if every proper submodule of M is contained in a maximal
submodule of M [12].
Corollary 3.6. Let M be a radical lifting module. Then
M
P (M)
is coatomic and \oplus -supplemented.
Proof. Since P (M) is a fully invariant submodule of M, it follows from Proposition 3.3 that
M
P (M)
is radical lifting. Applying Lemma 3.1, we obtain that
M
P (M)
is srs\oplus . By Lemma 3.3,
M
P (M)
has a small radical. Thus, it is \oplus -supplemented by [8] (Lemma 2.2).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
112 B. N. TÜRKMEN
It follows from [10] (42.1) that
M
P (M)
is coatomic.
Proposition 3.4. Let M be a radical lifting module and X be a submodule with \mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq X.
If X is a direct summand of M, then X is a radical lifting module.
Proof. Let N be any submodule of X such that \mathrm{R}\mathrm{a}\mathrm{d}(X) \subseteq N. By [8] (Lemma 2.28), we
obtain that \mathrm{R}\mathrm{a}\mathrm{d}(M) = \mathrm{R}\mathrm{a}\mathrm{d}(X). Since M is a radical lifting module, there exists a direct summand
N \prime of M such that N \prime \leq N, M = N \prime \oplus K for a submodule K \leq M, and N \cap K \ll K. Then
X = N+(X\cap K). Since M = N \prime \oplus K, then N = N \prime \oplus (X\cap K). It follows that N\cap (X\cap K) \ll N\cap K.
Therefore N is a radical lifting module.
References
1. Bass H. Finistic dimension and a homological generalization of semiprimary rings // Trans. Amer. Math. Soc. –
1960. – 95. – P. 466 – 488.
2. Büyükaşık E., Türkmen E. Strongly radical supplemented modules // Ukr. Math. J. – 2012. – 64, № 8. – P. 1306 – 1313.
3. Clark J., Lomp C., Vanaja N., Wisbauer R. Lifting modules. Supplements and projectivity in module theory // Front.
Math. – Basel: Birkhäuser, 2006.
4. Çalışıcı H., Pancar A. Cofinitely semiperfect modules // Sib. Math. J. – 2005. – 46, № 2. – P. 359 – 363.
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7. Mohamed S. H., Müller B. J. Continuous and discrete modules // London Math. Soc. – 1990. – 147.
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12. Zöschinger H. Komplementierte moduln über Dedekindringen // J. Algebra. – 1974. – 29. – P. 42 – 56.
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Received 04.11.13,
after revision — 28.10.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 1
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| id | umjimathkievua-article-1679 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:29Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2f/df14616e03aafd31457b5bafd1713c2f.pdf |
| spelling | umjimathkievua-article-16792019-12-05T09:23:35Z A generalization of semiperfect modules Узагальнення напiвдосконалих модулiв Türkmen, B. N. Тюркмен, Б. Н. A module $M$ is called radical semiperfect, if $\frac MN$ has a projective cover whenever $\mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M$. We study various properties of these modules. It is proved that every left $R$-module is radical semiperfect if and only if $R$ is left perfect. Moreover, radical lifting modules are defined as a generalization of lifting modules. Модуль $M$ називається радикальним напiвдосконалим, якщо $\frac MN$ має проективне покриття, як тiльки $\mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M$. Дослiджено рiзнi властивостi цих модулiв. Доведено, що кожен лiвий $R$-модуль є радикально напiвдосконалим тодi i тiльки тодi, коли $R$ є лiвим досконалим. Крiм того, радикальнi пiднiмаючi модулi визначено, як узагальнення пiднiмаючих модулiв. Institute of Mathematics, NAS of Ukraine 2017-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1679 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 1 (2017); 104-112 Український математичний журнал; Том 69 № 1 (2017); 104-112 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1679/661 Copyright (c) 2017 Türkmen B. N. |
| spellingShingle | Türkmen, B. N. Тюркмен, Б. Н. A generalization of semiperfect modules |
| title | A generalization of semiperfect modules |
| title_alt | Узагальнення напiвдосконалих модулiв |
| title_full | A generalization of semiperfect modules |
| title_fullStr | A generalization of semiperfect modules |
| title_full_unstemmed | A generalization of semiperfect modules |
| title_short | A generalization of semiperfect modules |
| title_sort | generalization of semiperfect modules |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1679 |
| work_keys_str_mv | AT turkmenbn ageneralizationofsemiperfectmodules AT tûrkmenbn ageneralizationofsemiperfectmodules AT turkmenbn uzagalʹnennânapivdoskonalihmoduliv AT tûrkmenbn uzagalʹnennânapivdoskonalihmoduliv AT turkmenbn generalizationofsemiperfectmodules AT tûrkmenbn generalizationofsemiperfectmodules |