Principally Goldie$\ast$ -lifting modules
A module $M$ is called а principal Goldie$\ast$ -lifting if, for every proper cyclic submodule $X$ of $M$, there is a direct summand $D$ of $M$ such that $X\beta \ast D$. We focus our attention on principally Goldie $\ast$ -lifting modules as a generalization of lifting modules. Various properties...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507418815889408 |
|---|---|
| author | Guroglu, A. T. Meric, E. T. Гюрогли, А. Т. Мерис, Е. Т. |
| author_facet | Guroglu, A. T. Meric, E. T. Гюрогли, А. Т. Мерис, Е. Т. |
| author_sort | Guroglu, A. T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:20:38Z |
| description | A module $M$ is called а principal Goldie$\ast$ -lifting if, for every proper cyclic submodule $X$ of $M$, there is a direct summand
$D$ of $M$ such that $X\beta \ast D$. We focus our attention on principally Goldie $\ast$ -lifting modules as a generalization of lifting
modules. Various properties of these modules are presented. |
| first_indexed | 2026-03-24T02:09:00Z |
| format | Article |
| fulltext |
UDC 512.5
A. T. Güroğlu, E. T. Meriç (Celal Bayar Univ., Manisa, Turkey)
PRINCIPALLY GOLDIE\ast -LIFTING MODULES
ГОЛОВНI ГОЛДI\ast -ЛIФТИНГ МОДУЛI
A module M is called а principal Goldie\ast -lifting if, for every proper cyclic submodule X of M, there is a direct summand
D of M such that X\beta \ast D. We focus our attention on principally Goldie\ast -lifting modules as a generalization of lifting
modules. Various properties of these modules are presented.
Модуль називається головним Голдi\ast -лiфтингом, якщо для кожного власного циклiчного субмодуля X модуля M
iснує прямий доданок D з M такий, що X\beta \ast D. Ми зосереджуємо нашу увагу на головних Голдi\ast -лiфтинг модулях,
що розглядаються як узагальнення лiфтинг модулiв. Наведено рiзнi властивостi таких модулiв.
1. Introduction. Throughout this paper, R denotes an associative ring with identity and all modules
are unital right R-modules. \mathrm{R}\mathrm{a}\mathrm{d}(M) will denote the Jacobson radical of M. Let M be an R-
module and N,K be submodules of M. The submodule K of M will be denoted by K \leq M. K
is called small (or superfluous) in M, denoted by K \ll M, if, for every submodule N of M, the
equality K + N = M implies N = M. K is called a supplement of N in M if K is minimal
with respect to N +K = M, equivalently K +N = M and K \cap N \ll K. A module M is called
supplemented (weakly supplemented) if every submodule of M has a supplement (weak supplement)
in M. A module M is \oplus -supplemented if every submodule of M has a supplement which is a direct
summand of M. [1] defines principally supplemented modules and investigates their properties. A
module M is said to be principally supplemented if for all cyclic submodule X of M there exists
a submodule N of M such that M = N + X and N \cap X \ll N. A module M is said to be
\oplus -principally supplemented if, for each cyclic submodule X of M, there exists a direct summand D
of M such that M = D +X and D \cap X \ll D. A nonzero module M is said to be hollow if every
proper submodule of M is small in M. A nonzero module M is said to be principally hollow if every
proper cyclic submodule of M is small in M. Clearly, hollow modules are principally hollow. Given
submodules K \subseteq N \subseteq M, the inclusion K \lhook \rightarrow N is called cosmall in M, denoted by K
cs
\lhook \rightarrow N, if
N/K \ll M/K.
Lifting modules play an important role in module theory. Also their various generalizations are
studied by many authors in [1, 2, 5 – 7, 9, 10]. A module M is called lifting if, for every submodule
N of M, there is a decomposition M = D \oplus D\prime such that D \subseteq N and D\prime \cap N \ll M. A module
M is called principally lifting if for all cyclic submodule X of M, there exists a decomposition
M = D \oplus D\prime such that D \subseteq X and D\prime \cap X \ll M. A module M is said to be H -supplemented
if, for every submodule N, there is a direct summand D of M such that M = N + B holds if and
only if M = D + B for any submodule B of M. G. F. Birkenmeier et al. [2] defines \beta \ast relation
to study on the open problem ‘Is every H -supplemented module supplemented?’ in [7]. They say
submodules X, Y of M are \beta \ast equivalent, X\beta \ast Y, if and only if
X + Y
X
is small in
M
X
and
X + Y
Y
is small in
M
Y
. M is called Goldie\ast -lifting (or briefly, \scrG \ast -lifting) if and only if for each X \leq M,
there exists a direct summand D of M such that X\beta \ast D. M is called Goldie\ast -supplemented (or
c\bigcirc A. T. GÜROĞLU, E. T. MERIÇ, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7 905
906 A. T. GÜROĞLU, E. T. MERIÇ
briefly, \scrG \ast -supplemented) if and only if for each X \leq M, there exists a supplement submodule S
of M such that X\beta \ast S (see [2]).
Section 2 is based on principally Goldie\ast -lifting modules. These modules are considered as gene-
ralization of Goldie\ast -lifting modules. We give some necessary assumptions for a factor module or a
direct summand of a principally Goldie\ast -lifting module to be principally Goldie\ast -lifting. Principally
lifting, principally Goldie\ast -lifting and principally supplemented modules are compared. Finally, we
show that principally lifting, principally Goldie\ast -lifting and \oplus -principally supplemented coincide
on \pi -projective modules. In addition, one of the our aims is to determine the connection between
principally Goldie\ast -lifting and Goldie\ast -lifting. As a consequence, we prove this relation under some
restriction.
2. Principally Goldie\ast -lifting modules. In [2], G. F. Birkenmeier et al. defined \beta \ast relation.
We start this section by giving some properties of \beta \ast relation without proofs. The proofs of the
following notions can be found in [2]. Moreover, in [2], the authors introduced two notions called
Goldie\ast -supplemented module and Goldie\ast -lifting module depend on the \beta \ast relation. They showed
that Goldie\ast -lifting modules and H -supplemented modules coincide in [2] (Theorem 3.6). In this
section, we define principally Goldie\ast -lifting module (briefly principally \scrG \ast -lifting module) as a
generalization of \scrG \ast -lifting module and investigate some properties of this module. In particular, we
prove that principally \scrG \ast -lifting and \scrG \ast -lifting coincide under some conditions.
Definition 2.1 ([2], Definition 2.1). Any submodules X,Y of M are \beta \ast equivalent, X\beta \ast Y, if
and only if
X + Y
X
is small in
M
X
and
X + Y
Y
is small in
M
Y
.
Lemma 2.1 ([2], Lemma 2.2). \beta \ast is an equivalence relation.
By [2, p. 43], the zero submodule is \beta \ast equivalent to any small submodule.
Theorem 2.1 ([2], Theorem 2.3). Let X,Y be submodules of M. The following are equivalent:
(a) X\beta \ast Y ;
(b) X
cs
\lhook \rightarrow X + Y and Y
cs
\lhook \rightarrow X + Y ;
(c) for each submodule A of M such that X+Y +A = M, then X+A = M and Y +A = M ;
(d) if K \leq M with X +K = M, then Y +K = M, and if H \leq M with Y +H = M, then
X +H = M.
Theorem 2.2 ([2], Theorem 2.6). Let X,Y be submodules of M such that X\beta \ast Y.Then
1) X \ll M if and only if Y \ll M ;
2) X has a (weak) supplement C in M if and only if C is a (weak) supplement for Y.
Lemma 2.2. Let M = D \oplus D\prime and A,B \leq D. Then A\beta \ast B in M if and only if A\beta \ast B in D.
Proof. (\Rightarrow ) Let A\beta \ast B in M and A+B +N = D for some submodule N of D. Let us show
that A+N = D and B +N = D. Since A\beta \ast B in M,
M = D \oplus D\prime = A+B +N +D\prime
implies A+N +D\prime = M and B +N +D\prime = M. By [11, p. 41], A+N = D and B +N = D.
From Theorem 2.1, we get A\beta \ast B in D.
(\Leftarrow ) Let A\beta \ast B in D. Then
A+B
A
\ll D
A
implies
A+B
A
\ll M
A
. Similarly,
A+B
B
\ll D
B
implies
A+B
B
\ll M
B
. This means that A\beta \ast B in M.
Lemma 2.3. If a direct summand D of M is \beta \ast equivalent to a cyclic submodule X of M,
then D is also cyclic.
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PRINCIPALLY GOLDIE\ast -LIFTING MODULES 907
Proof. Assume that M = D \oplus D\prime for some submodules D,D\prime of M and X is a cyclic
submodule of M which is \beta \ast equivalent to D. By Theorem 2.1 (c), M = X+D\prime . Since
X +D\prime
D\prime =
=
M
D\prime
\sim = D and X is cyclic, D is cyclic.
Definition 2.2. A module M is called principally Goldie\ast -lifting (briefly principally \scrG \ast -lifting)
if for each cyclic submodule X of M, there exists a direct summand D of M such that X\beta \ast D.
Clearly, every \scrG \ast -lifting module is principally \scrG \ast -lifting. However, the converse does not hold
as the next example shows.
Example 2.1. Consider the \BbbZ -module \BbbQ . Since \mathrm{R}\mathrm{a}\mathrm{d}(\BbbQ ) = \BbbQ , every cyclic submodule of \BbbQ is
small in \BbbQ . By [2] (Example 2.15), the \BbbZ -module \BbbQ is principally \scrG \ast -lifting. But the \BbbZ -module \BbbQ
is not supplemented. It follows from [2] (Theorem 3.6) that it is not \scrG \ast -lifting.
A module M is said to be radical if \mathrm{R}\mathrm{a}\mathrm{d}(M) = M.
Lemma 2.4. Every radical module is principally \scrG \ast -lifting.
Proof. Let m \in M. As M is radical, mR \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(M). By [11] (21.5), mR \ll M. So we get
mR\beta \ast 0. Thus M is principally \scrG \ast -lifting.
Theorem 2.3. Let M be a module. Consider the following conditions:
(a) M is principally lifting,
(b) M is principally \scrG \ast -lifting,
(c) M is principally supplemented.
Then (a)\Rightarrow (b) \Rightarrow (c).
Proof. (a) \Rightarrow (b) Let m \in M. From (a), there is a decomposition M = D \oplus D\prime such that
D \leq mR and mR \cap D\prime \ll M. Since D \leq mR,
mR+D
mR
\ll M
mR
. By modularity, mR =
= M \cap mR = (D \oplus D\prime ) \cap mR = D \oplus (mR \cap D\prime ). Then
mR
D
\sim = mR \cap D\prime and
M
D
\sim = D\prime . If
mR \cap D\prime \ll M, by [11] (19.3), mR \cap D\prime \ll D\prime . It implies that
mR+D
D
\ll M
D
. Therefore it is
seen that mR\beta \ast D from Definition 2.1. Hence M is principally \scrG \ast -lifting.
(b) \Rightarrow (c) Let m \in M. By the hypothesis, there exists a direct summand D of M such that
mR\beta \ast D. Since M = D \oplus D\prime for some submodule D\prime of M and D\prime is a supplement of D, D\prime is a
supplement of mR in M by [2] (Theorem 2.6 (ii)). Thus M is principally supplemented.
We expect that a principally \scrG \ast -lifting module is principally lifting. But unfortunately, it is not
true in general:
Example 2.2. Consider the \BbbZ -module M = \BbbZ /2\BbbZ \oplus \BbbZ /8\BbbZ . From [10] (Example 3.7), we can
say that M is a H -supplemented module. Then M is \scrG \ast -lifting by [2] (Theorem 3.6). Since every
\scrG \ast -lifting module is principally \scrG \ast -lifting, M is also principally \scrG \ast -lifting. But from [1] (Exam-
ples 7.(3)), M is not principally lifting.
Theorem 2.4. Let M be an indecomposable module. Then the following conditions are equiva-
lent:
(a) M is principally lifting,
(b) M is principally hollow,
(c) M is principally \scrG \ast -lifting.
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908 A. T. GÜROĞLU, E. T. MERIÇ
Proof. (a) \leftrightarrow (b) It is easy to see from [1] (Lemma 14).
(b) \Rightarrow (c) Suppose that M is principally hollow and m \in M. Then mR \ll M. It means that
mR\beta \ast 0.
(c) \Rightarrow (b) Let mR be a proper cyclic submodule of M. By (c), there exists a decomposition
M = D \oplus D\prime such that mR\beta \ast D. Since M is indecomposable, D = M or D = 0. If D = M,
from [2] (Corollary 2.8 (iii)), we obtain that mR = M, which is a contradiction. Thus D must be
zero, that is, mR\beta \ast 0 and we have mR \ll M. Hence M is principally hollow.
We shall give the following example of modules which are principally supplemented but not
principally \scrG \ast -lifting.
Example 2.3. Let F be a field, x and y commuting indeterminates over F. Let R = F [x, y]
be a polynomial ring and I1 = (x2) and I2 = (y2) be ideals of R and the ring S = R/(x2, y2).
Consider the S -module M = xS + yS. By [1] (Example 15), M is an indecomposable S -module
and it is not principally hollow. Then from Theorem 2.4 M is not principally \scrG \ast -lifting. Therefore
it follows from [1] (Example 15) that M is principally supplemented.
A module M is said to be principally semisimple if every cyclic submodule of M is a direct
summand of M.
Lemma 2.5. Every principally semisimple module is principally \scrG \ast -lifting.
Proof. Let X be a cyclic submodule of M. By the assumption, X is a direct summand of M.
Then M = X \oplus X \prime for some submodule X \prime of M. Since \beta \ast is an equivalence relation, we have
X\beta \ast X. Thus M is principally \scrG \ast -lifting.
Recall that a submodule N of M is called fully invariant if for each endomorphism f of M,
f(N) is contained in N. Clearly 0 and M are fully invariant submodules of M. A module M is
said to be a duo module provided every submodule of M is fully invariant. For example, if M is
a simple right R-module, then M is a duo module but M \oplus M is not duo (see [8]). A module M
is called distributive if for all submodules A,B,C of M, A + (B \cap C) = (A + B) \cap (A + C) or
A \cap (B + C) = (A \cap B) + (A \cap C) (see [3]).
Proposition 2.1. Let M = M1 \oplus M2 be a duo module (or distributive module). Then M is
principally \scrG \ast -lifting if and only if M1 and M2 are principally \scrG \ast -lifting.
Proof. (\Rightarrow ) Take any m \in M1. Since M is principally \scrG \ast -lifting, then for m \in M, there
exists a direct decomposition M = D \oplus D\prime such that mR\beta \ast D in M for D,D\prime \leq M. As M is a
duo module, it is obtained that M1 = (M1 \cap D) \oplus (M1 \cap D\prime ). We claim that mR\beta \ast (M1 \cap D) in
M1. To prove this, it is enough to show that for some submodule A of M1, M1 = mR + A and
M1 = (M1 \cap D) +A. Let M1 = mR+ (M1 \cap D) +A for some submodule A of M1. Then
M = M1 \oplus M2 = [mR+ (M1 \cap D) +A]\oplus M2 = mR+D +A+M2.
By Theorem 2.1, M = D + A + M2 and M = mR + A + M2. Because M is duo, we can
write as M1 = M1 \cap (D + A + M2) = A + [M1 \cap (D + M2)] = A + (M1 \cap D) and M1 =
= M1\cap (mR+A+M2) = mR+A. Again by Theorem 2.1, we get mR\beta \ast (M1\cap D) in M1. Hence
M1 is principally \scrG \ast -lifting. Similarly, it can be showed that M2 is principally \scrG \ast -lifting.
(\Leftarrow ) Let m \in M. If M is a duo module, for the cyclic submodule mR of M, mR = (mR \cap
\cap M1) \oplus (mR \cap M2). If M = M1 \oplus M2, then mR = m1R + m2R for some m1 \in M1 and
m2 \in M2. So mR \cap M1 = m1R and mR \cap M2 = m2R. Since M1 and M2 are principally \scrG \ast -
lifting, there are decompositions M1 = D1 \oplus D\prime
1 and M2 = D2 \oplus D\prime
2 such that m1R\beta \ast D1 in M1
and m2R\beta \ast D2 in M2. By Lemma 2.2, m1R\beta \ast D1 and m2R\beta \ast D2 in M. By [2] (Proposition 2.11),
(m1R+m2R)\beta \ast (D1 \oplus D2). Since mR = m1R+m2R, we get mR\beta *(D1 \oplus D2).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
PRINCIPALLY GOLDIE\ast -LIFTING MODULES 909
Let M, N and P be R-modules. P is called M -projective if for each epimorphism g : M \rightarrow N
and each homomorphism f : P \rightarrow N, there exists a homomorphism h : P \rightarrow M such that gh = f.
If P is P -projective, then P is also called self-injective (or quasi-injective). An R-module M is
said to be \pi -projective if for every two submodules U, V of M with U + V = M there exists
f \in \mathrm{E}\mathrm{n}\mathrm{d}(M) with \mathrm{I}\mathrm{m}(f) \subset U and \mathrm{I}\mathrm{m}(1 - f) \subset V. Clearly every self-projective module is also
\pi -projective [11].
Proposition 2.2. Let any cyclic submodule of M have a supplement which is a relatively pro-
jective direct summand of M. Then M is principally \scrG \ast -lifting.
Proof. Let m \in M. By the hypothesis, there exists a decomposition
M = D \oplus D\prime such that M = mR + D\prime and mR \cap D\prime \ll D\prime . Because D is D\prime -projective,
M = A \oplus D\prime for some submodule A of mR by [7] (Lemma 4.47). So M is principally lifting. It
follows from Theorem 2.3 that M is principally \scrG \ast -lifting.
Proposition 2.3. Let M be principally \scrG \ast -lifting and N be a submodule of M. If
N +D
N
is a
direct summand of
M
N
for any cyclic direct summand D of M, then
M
N
is principally \scrG \ast -lifting.
Proof. Let
mR+N
N
be a cyclic submodule of
M
N
for m \in M. If M is principally \scrG \ast -lifting,
there exists a decomposition M = D\oplus D\prime such that mR\beta \ast D. Then D is also cyclic from Lemma 2.3.
By the hypothesis,
D +N
N
is a direct summand in
M
N
. We claim that
mR+N
N
\beta \ast D +N
N
. Consider
the canonical epimorphism \theta : M \rightarrow M/N. By [2] (Proposition 2.9 (i)), \theta (mR)\beta \ast \theta (D), that is,
mR+N
N
\beta \ast D +N
N
. Thus
M
N
is principally \scrG \ast -lifting.
Corollary 2.1. Let M be principally \scrG \ast -lifting. Then
(a) If M is a distributive (or duo) module, then any factor module of M is principally \scrG \ast -lifting.
(b) Let N be a projection invariant, that is, eN \subseteq N for all e2 = e \in \mathrm{E}\mathrm{n}\mathrm{d}(M). Then
M
N
is
principally \scrG \ast -lifting. In particular,
M
A
is principally \scrG \ast -lifting for every fully invariant submodule
A of M.
Proof. (a) Let N be any submodule of M and D be a cyclic direct summand of M. Note that
M = D \oplus D\prime for some submodules D\prime of M. Therefore we have
M
N
=
D \oplus D\prime
N
=
D +N
N
+
D\prime +N
N
.
We will show that
D +N
N
\cap D\prime +N
N
= 0. Since M is distributive and D \cap D\prime = 0,
(D +N) \cap (D\prime +N) =
\bigl(
(D +N) \cap D\prime \bigr) + \bigl(
(D +N) \cap N
\bigr)
= (D \cap D\prime ) + (N \cap D\prime ) +N = N.
We obtain
M
N
=
D +N
N
\oplus D\prime +N
N
. By Proposition 2.3,
M
N
is principally \scrG \ast -lifting.
(b) Let D be a cyclic direct summand of M and N be a projection invariant of M. Then
M = D \oplus D\prime for some D\prime \leq M. For the projection map \pi D : M \rightarrow D, \pi 2
D = \pi \in \mathrm{E}\mathrm{n}\mathrm{d}(M) and
\pi D(N) \subseteq N. So \pi D(N) = N \cap D. Similarly, \pi D\prime (N) = N \cap D\prime for the projection map \pi D\prime :
M \rightarrow D\prime . Hence we have N = (N \cap D) + (N \cap D\prime ). So
M = (D +N) + (D\prime +N) =
\bigl[
D + (N \cap D) + (N \cap D\prime )
\bigr]
+ (D\prime +N) =
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
910 A. T. GÜROĞLU, E. T. MERIÇ
=
\bigl[
D \oplus (N \cap D\prime )
\bigr]
+ (D\prime +N)
and, by modularity,\bigl[
D \oplus (N \cap D\prime )
\bigr]
\cap (D\prime +N) =
\bigl[
D \cap (D\prime +N)
\bigr]
+ (N \cap D\prime ) = (N \cap D) + (N \cap D\prime ) = N.
Thus it can be seen that
M
N
=
D \oplus (N \cap D\prime )
N
\oplus D\prime +N
N
. By Proposition 2.3,
M
N
is principally
\scrG \ast -lifting.
Another consequence of Proposition 2.2 is given in the next result.
A module M is said to have the summand sum property (SSP) if the sum of any two direct
summands of M is again a direct summand.
Proposition 2.4. Let M be a principally \scrG \ast -lifting module. If M has SSP, then any direct
summand of M is principally \scrG \ast -lifting.
Proof. Let M = N \oplus N \prime for some submodules N,N \prime of M. Our aim is to show that N is
principally \scrG \ast -lifting. Take any cyclic direct summand D of M. From the SSP property, we can
write as M = (D +N \prime )\oplus T for some submodule T of M. Then
N \sim =
M
N \prime =
D +N \prime
N \prime +
T +N \prime
N \prime .
By modular law,
(D +N \prime ) \cap (T +N \prime ) = N \prime +
\bigl[
(D +N \prime ) \cap T
\bigr]
= N \prime .
So we obtain
M
N \prime =
D +N \prime
N \prime \oplus T +N \prime
N \prime .
Using Proposition 2.3, it can be said that N \sim =
M
N \prime is principally \scrG \ast -lifting.
Next, we give a sufficient condition for M/\mathrm{R}\mathrm{a}\mathrm{d}(M) is principally semisimple in case M is
principally \scrG \ast -lifting module.
Proposition 2.5. Let M be principally \scrG \ast -lifting and distributive module. Then
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
is
principally semisimple.
Proof. Let m \in M. By the assumption, there exists a decomposition M = D \oplus D\prime such that
mR\beta *D for some submodule D,D\prime of M. By [2] (Theorem 2.6 (ii)), D\prime is a supplement of mR,
that is, M = mR+D\prime and mR \cap D\prime \ll D\prime . Then
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
=
mR+D\prime
\mathrm{R}\mathrm{a}\mathrm{d}(M)
=
mR+\mathrm{R}\mathrm{a}\mathrm{d}(M)
\mathrm{R}\mathrm{a}\mathrm{d}(M)
+
D\prime +\mathrm{R}\mathrm{a}\mathrm{d}(M)
\mathrm{R}\mathrm{a}\mathrm{d}(M)
.
Because M is distributive,\bigl(
mR+\mathrm{R}\mathrm{a}\mathrm{d}(M)
\bigr)
\cap
\bigl(
D\prime +\mathrm{R}\mathrm{a}\mathrm{d}(M)
\bigr)
= (mR \cap D\prime ) + \mathrm{R}\mathrm{a}\mathrm{d}(M).
Since mR\cap D\prime \ll D\prime , so mR\cap D\prime \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(M). In this case,
\bigl(
mR+\mathrm{R}\mathrm{a}\mathrm{d}(M)
\bigr)
\cap
\bigl(
D\prime +\mathrm{R}\mathrm{a}\mathrm{d}(M)
\bigr)
=
= \mathrm{R}\mathrm{a}\mathrm{d}(M). As a result,
mR+\mathrm{R}\mathrm{a}\mathrm{d}(M)
\mathrm{R}\mathrm{a}\mathrm{d}(M)
is a direct summand in
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
, this means that
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
is a principally semisimple module.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
PRINCIPALLY GOLDIE\ast -LIFTING MODULES 911
Proposition 2.6. Let M be a principally \scrG \ast -lifting module and \mathrm{R}\mathrm{a}\mathrm{d}(M) \ll M. Then
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
is principally semisimple.
Proof. Let
X
\mathrm{R}\mathrm{a}\mathrm{d}(M)
be a cyclic submodule of
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
for any submodule X of M containing
\mathrm{R}\mathrm{a}\mathrm{d}(M). Then X = mR + \mathrm{R}\mathrm{a}\mathrm{d}(M) for some m \in M. By the assumption, there
exists a decomposition M = D \oplus D\prime such that mR\beta \ast D for submodules D,D\prime \leq M. It fol-
lows from [2] (Corollary 2.12) that
\bigl(
mR + \mathrm{R}\mathrm{a}\mathrm{d}(M)
\bigr)
\beta \ast D. Moreover, D\prime is a supplement of
mR + \mathrm{R}\mathrm{a}\mathrm{d}(M) in M from by [2] (Theorem 2.6 (ii)). Then we have M = mR + \mathrm{R}\mathrm{a}\mathrm{d}(M) + D\prime
and D\prime \cap
\bigl(
mR+\mathrm{R}\mathrm{a}\mathrm{d}(M)
\bigr)
= D\prime \cap X \ll D\prime , that is, D\prime \cap X \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(M). On the other hand,
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
=
X
\mathrm{R}\mathrm{a}\mathrm{d}(M)
+
D\prime +\mathrm{R}\mathrm{a}\mathrm{d}(M)
\mathrm{R}\mathrm{a}\mathrm{d}(M)
.
By modular law,
X
\mathrm{R}\mathrm{a}\mathrm{d}(M)
\cap D\prime +\mathrm{R}\mathrm{a}\mathrm{d}(M)
\mathrm{R}\mathrm{a}\mathrm{d}(M)
=
(X \cap D\prime ) + \mathrm{R}\mathrm{a}\mathrm{d}(M)
\mathrm{R}\mathrm{a}\mathrm{d}(M)
and since X \cap D\prime \subseteq \mathrm{R}\mathrm{a}\mathrm{d}(M), we obtain
M
\mathrm{R}\mathrm{a}\mathrm{d}(M)
=
X
\mathrm{R}\mathrm{a}\mathrm{d}(M)
\oplus D\prime +\mathrm{R}\mathrm{a}\mathrm{d}(M)
\mathrm{R}\mathrm{a}\mathrm{d}(M)
.
Theorem 2.5 ([4], 4.14). Let M be \pi -projective and let U, V \leq M be submodules with M =
= U + V.
(1) If U is a direct summand in M, then there exists V \prime \subset V with M = U \oplus V \prime .
(2) If U \cap V = 0, then V is U -projective (and U is V -projective).
(3) If U \cap V = 0 and V \sim = U, then M is self-projective.
(4) If U and V are direct summands of M, then U \cap V is also direct summand of M.
In general, it is not true that principally lifting and principally \scrG \ast -lifting modules coincide. As
we will see in the following theorem, we need \pi -projectivity condition.
Theorem 2.6. Let M be a module. Consider the following conditions:
(a) M is principally lifting,
(b) M is principally \scrG \ast -lifting,
(c) M is \oplus -principally supplemented.
Then (a) \Rightarrow (b) \Rightarrow (c). If M is \pi -projective, then (c)\Rightarrow (a) holds.
Proof. (a) \Rightarrow (b) It follows from Theorem 2.3.
(b) \Rightarrow (c) It follows from [2] (Theorem 2.6 (ii)).
(c) \Rightarrow (a) Consider any m \in M. By the assumption, mR has a supplement D which is a direct
summand in M, that is, M = mR+D = D\oplus A and mR \cap D \ll D for some submodule A of M.
Since M is \pi -projective, there exists a complement D\prime of D such that D\prime \subseteq mR by [4] (4.14 (1)).
Then we have M = D \oplus D\prime . Thus M is principally lifting.
Proposition 2.7. Let M be a \pi -projective module. Then M is principally \scrG \ast -lifting if and only
if every cyclic submodule X of M can be written as X = D \oplus A such that D is a direct summand
in M and A \ll M.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
912 A. T. GÜROĞLU, E. T. MERIÇ
Proof. (\Rightarrow ) Suppose M is principally \scrG \ast -lifting and \pi -projective module. By Theorem 2.6, M
is principally lifting. Then we observe that for any cyclic submodule X of M, there exists a direct
decomposition M = D\oplus D\prime such that D \leq X and X \cap D\prime \ll M. By modularity, we conclude that
X = D \oplus (X \cap D\prime ).
(\Leftarrow ) Let X be any cyclic submodule of M. By the assumption and [5] (Lemma 2.10), M is
principally lifting. Therefore from Theorem 2.6, M is principally \scrG \ast -lifting.
Now we mention that principally \scrG \ast -lifting and \scrG \ast -lifting modules coincide under some condi-
tions.
Proposition 2.8. Let M be Noetherian and have SSP. Then M is principally \scrG \ast -lifting if and
only if M is \scrG \ast -lifting.
Proof. (\Leftarrow ) Clear.
(\Rightarrow ) If M is Noetherian, for any submodule X of M there exist some m1,m2, . . . ,mn \in M
such that X = m1R +m2R + . . . +mnR by [11] (27.1). Since M is principally \scrG \ast -lifting, there
exist some direct summands D1, D2, . . . , Dn of M such that m1R\beta \ast D1, m2R\beta \ast D2, . . . ,mnR\beta \ast Dn.
Then D = D1 +D2 + . . . +Dn is also a direct summand in M because of SSP. By [2] (Proposi-
tion 2.11), X\beta \ast D. Hence M is \scrG \ast -lifting.
Proposition 2.9. Let any submodule N of M be a sum of a cyclic submodule X and a small
submodule A in M. Then M is principally \scrG \ast -lifting if and only if M is \scrG \ast -lifting.
Proof. (\Leftarrow ) Clear.
(\Rightarrow ) Let N be any submodule of M and N = X + A for a cyclic submodule X and a small
submodule A of M. Since M is principally \scrG \ast -lifting, there exists a direct summand D of M such
that X\beta \ast D. From [2] (Corollary 2.12), (X +A)\beta \ast D, that is, N\beta \ast D. Hence M is \scrG \ast -lifting.
References
1. Acar U., Harmancı A. Principally supplemented modules // Alban. J. Math. – 2010. – 4, № 3. – P. 79 – 88.
2. Birkenmeier G. F., Mutlu F. T., Nebiyev C., Sokmez N., Tercan A. Goldie*-supplemented modules // Glasg. Math. J. –
2010. – 52. – P. 41 – 52.
3. Camillo V. Distributive modules // J. Algebra. – 1975. – 36, № 1. – P. 16 – 25.
4. Clark J., Lomp C., Vanaja N., Wisbauer R. Lifting modules: supplements and projectivity in module theory. – Basel,
Switzerland: Birkhäuser-Verlag, 2006.
5. Kamal M., Yousef A. On principally lifting modules // Int. Electron. J. Algebra. – 2007. – 2. – P. 127 – 137.
6. Koşan T., Tütüncü Keskin D. H -supplemented duo modules // J. Algebra and Appl. – 2007. – 6, Issue 6. – P. 965 – 971.
7. Mohamed S. H., Müller B. J. Continuous and discrete modules // London Math. Soc. Lecture Note Ser. – 1990. –
147.
8. Özcan A.Ç., Harmancı A. Duo modules // Glasg. Math. J. – 2006. – 48. – P. 533 – 545.
9. Talebi Y., Hamzekolaee A. R., Tercan A. Goldie-rad-supplemented modules // An. Ştiinţ. Univ. “Ovidius” Constanţa.
Ser. Mat. – 2014. – 22, № 3. – P. 205 – 218.
10. Yongduo W., Dejun W. On H -supplemented modules // Communs Algebra. – 2012. – 40. – P. 3679 – 3689.
11. Wisbauer R. Foundations of module and ring theory. – Gordon and Breach, 1991.
Received 27.02.14,
after revision — 18.03.18
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
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| id | umjimathkievua-article-1604 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:00Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5f/0c221098b898c2a08221d720bc34125f.pdf |
| spelling | umjimathkievua-article-16042019-12-05T09:20:38Z Principally Goldie$\ast$ -lifting modules Головнi Голдi$\ast$ - лiфтинг модулi Guroglu, A. T. Meric, E. T. Гюрогли, А. Т. Мерис, Е. Т. A module $M$ is called а principal Goldie$\ast$ -lifting if, for every proper cyclic submodule $X$ of $M$, there is a direct summand $D$ of $M$ such that $X\beta \ast D$. We focus our attention on principally Goldie $\ast$ -lifting modules as a generalization of lifting modules. Various properties of these modules are presented. Модуль називається головним Голдi$\ast$ -лiфтингом, якщо для кожного власного циклiчного субмодуля $X$ модуля $M$ iснує прямий доданок $D$ з $M$ такий, що $X\beta \ast D$. Ми зосереджуємо нашу увагу на головних Голдi$\ast$ -лiфтинг модулях, що розглядаються як узагальнення лiфтинг модулiв. Наведено рiзнi властивостi таких модулiв. Institute of Mathematics, NAS of Ukraine 2018-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1604 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 7 (2018); 905-912 Український математичний журнал; Том 70 № 7 (2018); 905-912 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1604/586 Copyright (c) 2018 Guroglu A. T.; Meric E. T. |
| spellingShingle | Guroglu, A. T. Meric, E. T. Гюрогли, А. Т. Мерис, Е. Т. Principally Goldie$\ast$ -lifting modules |
| title | Principally Goldie$\ast$ -lifting modules |
| title_alt | Головнi Голдi$\ast$ - лiфтинг модулi |
| title_full | Principally Goldie$\ast$ -lifting modules |
| title_fullStr | Principally Goldie$\ast$ -lifting modules |
| title_full_unstemmed | Principally Goldie$\ast$ -lifting modules |
| title_short | Principally Goldie$\ast$ -lifting modules |
| title_sort | principally goldie$\ast$ -lifting modules |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1604 |
| work_keys_str_mv | AT gurogluat principallygoldieastliftingmodules AT mericet principallygoldieastliftingmodules AT gûrogliat principallygoldieastliftingmodules AT meriset principallygoldieastliftingmodules AT gurogluat golovnigoldiastliftingmoduli AT mericet golovnigoldiastliftingmoduli AT gûrogliat golovnigoldiastliftingmoduli AT meriset golovnigoldiastliftingmoduli |