On a class of $\lambda$ -modules
UDC 512.5 Smith in paper [{\it Mapping between module lattices}, Int. Electron. J. Algebra, {\bf 15}, 173–195 (2014)] introduced maps between the lattice of ideals of a commutative ring and the lattice of submodules of an $R$-module $M,$ i.e., $\mu$ and $\lambda$ mappings. ...
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| author | Wijayanti, I. E. Ardiyansyah, M. Prasetyo, P. W. I. E. Prasetyo, P. W. Wijayanti, I. E. Ardiyansyah, M. Prasetyo, P. W. |
| author_facet | Wijayanti, I. E. Ardiyansyah, M. Prasetyo, P. W. I. E. Prasetyo, P. W. Wijayanti, I. E. Ardiyansyah, M. Prasetyo, P. W. |
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| description | UDC 512.5
Smith in paper [{\it Mapping between module lattices}, Int. Electron. J. Algebra, {\bf 15}, 173–195 (2014)] introduced maps between the lattice of ideals of a commutative ring and the lattice of submodules of an $R$-module $M,$ i.e., $\mu$ and $\lambda$ mappings. The definitions of the maps were motivated by the definition of multiplication modules. Moreover, some sufficient conditions for the maps to be a lattice homomorphisms are studied. In this work we define a class of $\lambda$-modules and observe the properties of the class. We give a sufficient conditions for the module and the ring such that the class $\lambda$ is a hereditary pretorsion class.
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| doi_str_mv | 10.37863/umzh.v73i3.513 |
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DOI: 10.37863/umzh.v73i3.513
UDC 512.5
I. E. Wijayanti (Univ. Gadjah Mada, Indonesia),
M. Ardiyansyah (Aalto Univ., Finland),
P. W. Prasetyo (Univ. Ahmad Dahlan, Indonesia)
ON A CLASS OF \bfitlambda -MODULES
ПРО ОДИН КЛАС \bfitlambda -МОДУЛIВ
Smith in paper [Mapping between module lattices, Int. Electron. J. Algebra, 15, 173 – 195 (2014)] introduced maps between
the lattice of ideals of a commutative ring and the lattice of submodules of an R-module M, i.e., \mu and \lambda mappings. The
definitions of the maps were motivated by the definition of multiplication modules. Moreover, some sufficient conditions
for the maps to be a lattice homomorphisms are studied. In this work we define a class of \lambda -modules and observe the
properties of the class. We give a sufficient conditions for the module and the ring such that the class \lambda is a hereditary
pretorsion class.
У роботi [Mapping between module lattices, Int. Electron. J. Algebra, 15, 173 – 195 (2014)] Смiт увiв у розгляд вiдобра-
ження мiж решiткою iдеалiв комутативного кiльця та решiткою субмодулiв R-модуля M, тобто вiдображення \mu i \lambda .
Цi означення були мотивованi означеннями мультиплiкативних модулiв. Також було вказано деякi достатнi умови,
за яких цi вiдображення є гомоморфiзмами решiток. У цiй роботi наведено означення класу \lambda -модулiв та зазначено
властивостi цього класу. Вказано достатнi умови на модуль та кiльце, за яких клас \lambda є спадковим преторсiйним
класом.
1. Preliminaries. By the ring R we mean any commutative ring with unit and the module M means
a left R-module, except we state otherwise. An R-module M is called a multiplication module if
for any submodule N in M, there is an ideal I in R such that N = IM. For further explanation
of multiplication modules over commutative rings we refer to papers [4, 8, 13]. Moreover, M is a
multiplication module if and only if for any submodule N of M we have N = \mathrm{A}\mathrm{n}\mathrm{n}R(M/N)M
(see [8]).
An R-module M is called a prime module if for any non-zero submodule K in M, \mathrm{A}\mathrm{n}\mathrm{n}R(K) =
= \mathrm{A}\mathrm{n}\mathrm{n}R(M). A proper submodule N in M is called a prime submodule of M if M/N is a prime
module (see [14]).
Let K, N be submodules of M. The residue of K in N will be denoted by [N :R K] =
= \{ r \in R | rK \subseteq N\} . For a special case, that is if N = 0, we obtain the annihilator of K as
[0 :R K] = \mathrm{A}\mathrm{n}\mathrm{n}R(K).
Let \scrL (M) be the lattice of submodules of R-module M, where for any submodules N and K
in M the ’join’ and ’meet’ are defined as
N \vee K = N +K, N \wedge K = N \cap K,
and N \leq K means N \subseteq K. Especially, for M = R we have the lattice of ideals in R and it is
denoted by \scrL (R). The definition of \mu and \lambda mappings conducted by Smith in [12] are following:
\mu : \scrL (M) \rightarrow \scrL (R), N \mapsto \rightarrow \mathrm{A}\mathrm{n}\mathrm{n}R(M/N), (1.1)
\lambda : \scrL (R) \rightarrow \scrL (M), I \mapsto \rightarrow IM. (1.2)
c\bigcirc I. E. WIJAYANTI, M. ARDIYANSYAH, P. W. PRASETYO, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3 329
330 I. E. WIJAYANTI, M. ARDIYANSYAH, P. W. PRASETYO
The mappings (1.1) and (1.2) are motivated by the relationship of submodules and ideals in a multi-
plication module. Then we define a class of modules as following:
\lambda = \{ M | (B \cap C)M = BM \cap CM \forall B,C fi\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y} \mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{s} \mathrm{o}\mathrm{f} R\} .
Based on Lemma 2.1 of [12], M \in \lambda if and only if M is a \lambda -module. Note that \lambda is not necessary
a hereditary class.
If R is a ring, then an R-module M is called a chain module if for any submodules N and L in
M either N \subseteq L or L \subseteq N. The ring R is called a chain ring if the R-module R is a chain module.
Smith in Proposition 2.4 of [12] has proved a sufficient condition of a ring such that its modules are
in \lambda as follows:
Proposition 1.1. If the ring R is a chain ring, then every R-module is in \lambda .
Moreover, the class \lambda is closed under direct summands and direct sums (see Lemma 2.5 of [12]).
Theorem 2.3 of [12] gave a necessary and sufficient condition of a module to be in \lambda as we recall here.
Proposition 1.2. The following assertions are equivalent:
a) R is a Prüfer;
b) every R-module is in \lambda ;
c) the class \lambda is closed under the homomorphic image.
The sufficient conditions in Proposition 2.4 and Theorem 2.3 of [12] have given a motivation for
us to study more general situations from category R-modules R-Mod to subcategory \sigma [M ] which
consits M -subgenerated modules. In this work, we show that with some additional conditions, if
the subgenerator M is a Dedekind module or a chain module, then the class \lambda will be equal to the
class \sigma [M ].
In the next section, we discuss module Dedekind and the relationship with the class \lambda . In Section
3, we prove that we can generalize Theorem 2.3 of [12].
2. Dedekind modules and \bfitlambda -modules. For intensive study of Dedekind modules, we refer
to Alkan et al. [3] and Saraç et al. [11]. For any commutative ring R with identity and a set S
consisting of non-zero divisor elements of R, the fraction ring RS will be naturally formed. By
considering the notion of fractional ideals in Larsen and McCarthy [9], a fractional ideal I of R is
invertible if there exist a fractional ideal I - 1 of R such that I - 1I = R. In a case when I - 1 exists,
then I - 1 = [R :RS
I]. A domain R is called a Prüfer domain providing that each finitely generated
ideal of R is invertible. Furthermore, an integral domain R is Dedekind domain iff every non-zero
ideal of R is invertible.
Now we generalize the notion of invertibility of fractional ideals in the case of submodules.
Many papers have discussed the notion of invertible submodules (see, for example, [3, 11]).
For any R-module M, consider T = \{ t \in S | \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e} m \in M, tm = 0 \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s} m = 0\} .
We can see that T is a multiplicatively closed subset of S. For any submodule N of M, we define
N \prime = [M :RT
N ]. Following the concept of invertible ideal, we call a submodule N of M is
invertible if N \prime N = M. Then M is called a Dedekind module if every non-zero submodule of M
is invertible and M is called a Prüfer module providing every finitely generated non-zero submodule
is invertible. The examples of Dedekind modules are the \BbbZ -module \BbbQ and \BbbZ p for prime p.
An R-module M is called a multiplication module provided for each submodule N of M there
exist an ideal I of R such that N = IM, i.e., I = [N :R M ]. If P is a maximal ideal of R, then
we define
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON A CLASS OF \lambda -MODULES 331
TP (M) =
\bigl\{
m \in M | (1 - p)m = 0 for some p \in P
\bigr\}
. (2.1)
Next, M is P -cyclic if there exist p \in P and m \in M such that (1 - p)M \subseteq Rm. In El-Bast (2007)
it has been shown that M is multiplication module if and only if for every maximal ideal P of R
either M = TP (M) or M is P -cyclic.
Now we show the property of \lambda -module dealing with the invertibility property of submodules
of multiplication modules. Let us recall an important property in paper [1], that is for any finitely
generated faithful multiplication R-module and for any invertible submodule N of M, [N :R M ] is
an invertible ideal of R.
Proposition 2.1. Let M be an R-module. Then we have the following assertions:
1. If I is a multiplication ideal of a ring R and M is a multiplication R-module, then \lambda (I) is
a multiplication R-module.
2. Every invertible submodule N of a faithful multiplication finitely generated module M is a
\lambda -module.
3. If M is a faithful multiplication module over an integral domain R, then M is a \lambda -module
and for any ideal I of R, I - 1 = (\lambda (I)) - 1.
Proof. 1. Let P be a maximal ideal of R. Consider the set TP in (2.1). If TP (M) = M
or TP (I) = I, then TP (IM) = IM. Hence \lambda (I) is a multiplication module. Now suppose that
TP (I) \not = I and TP (M) \not = M. Then I and M are P -cyclic. Therefore there exist elements p1, p2 \in
\in P, a \in I, m \in M such that (1 - p1)I \subseteq Ra and (1 - p2)M \subseteq Rm. It follows that (1 - p)IM \subseteq
\subseteq R(am) where p = p1 + p2 - p1p2 \in P. Thus \lambda (I) = IM is P -cyclic. This proves that \lambda (I) is a
multiplication R-module.
2. According to Proposition 2.1 of [1], for any invertible submodule N of M, [N :R M ] is an
invertible ideal of R. By using (2), we can easily obtain that N = [N :R M ]M is a multiplication
R-module. If r \in \mathrm{A}\mathrm{n}\mathrm{n}R(N), then rN = 0 and hence rM = rN - 1N = 0. This implies r = 0.
Therefore, N is faihtful multiplication R-module. By using Theorem 2.12 of [12], we conclude that
N is a \lambda -module because every faithful multiplication module is \lambda -module.
3. It is obvious by Theorem 2.12 of [12] and Lemma 1 of [2].
Now we are ready to display the connection of Dedekind module with \lambda -module by using the
following Corollary 3.8 of [3]. We recall a module M is divisible if for any 0 \not = r \in R, M = rM.
Lemma 2.1. Let M be a Dedekind divisible R-module. Then R is a field.
It is easy to understand that any vector space is a \lambda -module. Moreover, we have the following
direct consequences of Lemma 2.1.
Proposition 2.2. If M is a Dedekind divisible R-module, then:
1) M is \lambda -module;
2) for any N \in \sigma [M ], N \in \lambda ;
3) the class of \lambda is closed under submodules and homomorphic images.
Now we apply a result in paper [1].
Lemma 2.2. If M is a faithful multiplication module, then M is a Dedekind (Prüfer) module if
and only if R is a Dedekind (Prüfer) domain.
Proposition 2.3. Let M be a faithful multiplication module and Prüfer. Then \sigma [M ] \subseteq \lambda .
Proof. By assumption and according to a result in Lemma 2.2 we obtain that R is a Prüfer
domain. Proposition 1.2 shows that \lambda is equal to the category of R-modules. It is clear that
\sigma [M ] \subseteq \Lambda .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
332 I. E. WIJAYANTI, M. ARDIYANSYAH, P. W. PRASETYO
For the converse, we give in the following corollary.
Corollary 2.1. Let R be a semisimple ring, M a faithful multiplication module and Prüfer. If M
is a subgenerator for any semisimple module, then \sigma [M ] = \lambda .
Proof. Applying Proposition 2.3, \sigma [M ] \subseteq \lambda . Now take any N \in \lambda . Since R is a semisimple
ring, N is also a semisimple module. Moreover, N \in \sigma [M ] and we prove \lambda \subseteq \sigma [M ] as well.
We recall a sufficient condition of a Dedekind module in Lemma 3.3 of [1].
Lemma 2.3. Let R be an integral domain and M a faithful multiplication module. If for any
non-zero prime submodule P of M is invertible, then M is Dedekind.
Proposition 2.4. Let R be an integral domain and M a faithful multiplication module. If for
any non-zero prime submodule P of M is invertible, then \sigma [M ] \subseteq \lambda .
Proof. It is obvious by applying Lemmas 2.3 and 2.2.
The following propositions are another properties of a Dedekind module.
Proposition 2.5. Let M be a faithful multiplication Dedekind module over an integral domain
R. If I is an ideal of R, then M is a \lambda -module over I and \lambda (I) is a \lambda -module over R.
Proof. Since I is an ideal of R, \lambda (I) = IM is a submodule of M. Hence IM is an invertible
submodule. We have I - 1 = (\lambda (I)) - 1 due to Proposition 2.1, i.e., I is an invertible ideal of R. By
using a result in [7], we conclude that I is a \lambda -module over R. For any ideal B and C of R we
have
(B \cap C)\lambda (I) = (B \cap C)IM = (BI \cap CI)M = BIM \cap CIM = B\lambda (I) \cap C\lambda (I).
This proves our assertion.
Now we give a sufficient condition of \lambda -module.
Proposition 2.6. Let M be a multiplication Dedekind R-module. Then every R-module is a
\lambda -module.
Proof. According to Theorem 3.12 of [3], a multiplication Dedekind R-module implies the ring
R is a Dedekind domain, i.e., a Prüfer domain. It means every R-module is a \lambda -module.
If M is an R/I -module, then under scalar multiplication am = (a + I)m,M becomes an R-
modules for every a \in R and m \in M. Conversely, if M is an R-module, then M is an A/I -module
with respect to (a+ I)m = am for every a+ I \in R/I and m \in M.
Proposition 2.7. Let R be a ring, M an R-module and I an ideal of R where I \subseteq [0 :R M ].
M is a \lambda -module over R if and only if M is a \lambda -module over R/I.
Proof. If M is a \lambda -module over R, then (B \cap C)M = BM \cap CM for every finitely generated
ideals B, C of R. Suppose B/I, C/I be any ideals of R. Then (B/I \cap C/I)M = ((B \cap C)/I)M.
Since (B\cap C)M = BM\cap CM, ((B\cap C)/I)M = (B/I)M\cap (C/I)M. This gives (B/I\cap C/I)M =
= (B/I)M \cap (C/I)M. So, M is a \lambda -module over R/I.
Conversely, let M is a \lambda -module over R/I. Suppose B, C be any ideals of R with BM \not = \{ 0\}
and CM \not = \{ 0\} . Clearly, B+I/I, C+I/I are an ideals of R/I. Since M is a \lambda -module over R/I,
((B + I/I)\cap (C + I/I))M = (B + I/I)M \cap (C + I/I)M. On the other hand, ((B + I/I)\cap (C +
+I/I))M = ((B+I\cap C+I)/I)M, consequently ((B+I\cap C+I)/I)M = (B+I/I)M\cap (C+I/I)M.
Then (B+ I \cap C + I)M = (B+ I)M \cap (C + I)M. Since, I \subseteq (0 : M), (B+ I)M \cap (C + I)M =
= BM \cap CM. So, M is a \lambda -module over R.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
ON A CLASS OF \lambda -MODULES 333
3. Chain modules and \bfitlambda -modules. In this section, we consider chain modules and the relation-
ship with \lambda -modules.
Proposition 3.1. Let M be a chain R-module and faithful. Then \sigma [M ] \subseteq \lambda .
Proof. For any N \in \sigma [M ], according to (15.1) of [15], N = \oplus \Lambda Rm\lambda , where m\lambda \in M (\BbbN ).
Since M is chain, N = Rm0 for some m0 \in M (\BbbN ). Now take any ideals B and C in the ring R.
We only have to prove that BN \cap CN \subseteq (B \cap C)N. Take any x \in BN \cap CN. Then x = bm0 for
some b \in B and x = cm0 for some c \in C. Hence x = bm0 = cm0 and moreover (b - c)m0 = 0.
Since M is faithful, b = c and we obtain that x \in (B \cap C)N.
For the converse of Proposition 3.1 we need an extra condition as we give in the next corollary.
Corollary 3.1. Let R be a semisimple ring, M a chain R-module, faithful and a subgenerator
for all semisimple R-modules. Then \sigma [M ] = \lambda .
Proof. We apply Proposition 3.1. It is known that a module over a semisimple ring is semisimple.
Take any R-module N in \lambda , then N is semisimple. From the assumption, N \in \sigma [M ].
According to the properties of \sigma [M ] in (15.1) of Wisbauer [15], we obtain the following corollary.
Corollary 3.2. Let R be a semisimple ring, M a chain R-module, faithful and a subgenerator
for all semisimple R-modules. Then:
1) \lambda is a hereditary pretorsion class, i.e., \lambda is closed under submodules, homomorphic images
and any direct sums;
2) for any N \in \lambda , N =
\sum
Rm, where m \in M (\BbbN );
3) pullback and pushout of morphisms in \lambda belong to \lambda .
Corollary 3.3. Let R be a semisimple ring, M a chain R-module, faithful and a subgenerator
for all semisimple R-modules. If N is M -injective, then N is K -injective for all K \lambda -module.
Proof. If N is M -injective, then N is K -injective for any K \in \sigma [M ]. But according to
Corollary 3.1, \sigma [M ] = \lambda . Hence N is K -injective for any K \in \lambda .
Now we recall the following definition from Definition 2.6 of [10].
Definition 3.1. Let M and N be R-modules. We says that M rises to N, denoted by M \uparrow N,
if every M -injective module is N -injective.
Based on this definition and properties of injectivity in \sigma [M ] we conclude that if N \in \sigma [M ],
then M \uparrow N, but the converse is not necessary true. Theorem 2.8 of [10] has given a sufficient
condition such that the converse is holds.
Corollary 3.4. Let R be a semisimple ring, M a chain R-module, faithful and a subgenerator
for all semisimple R-modules. For any module N which is M \uparrow N, N is M -injective if and only if
N is K -injective for all K \in \lambda .
Proof. It is straightforward from Corollary 3.3 and Theorem 2.8 of [10].
References
1. M. M. Ali, Invertibility of multiplication modules, New Zealand J. Math., 35, 17 – 29 (2006).
2. M. M. Ali, Invertibility of multiplication modules, II, New Zealand J. Math., 39, 45 – 64 (2009).
3. M. Alkan, B. Saraç, Y. Tiraş, Dedekind modules, Commun. Algebra, 33, 1617 – 1626 (2005).
4. R. Ameri, On the prime submodules of multiplication modules, Int. J. Math. and Math. Sci., 27, 1715 – 1724 (2003).
5. H. Ansari-Toroghy, F. Farshadifar, On multiplication and comultiplication modules, Acta Math. Sci. Ser. B., 31, № 2,
694 – 700 (2011).
6. S. Çeken, M. Alkan, P. F. Smith, The dual notion of the prime radical of a module, J. Algebra, 392, 265 – 275 (2013).
7. D. D. Anderson, On the ideal equation I(B \cap C) = IB \cap IC , Canad. Math. Bull., 26, № 3, 331 – 332 (1983).
8. Z. A. El-Bast, P. F. Smith, Multiplication modules, Commun. Algebra, 16, № 4, 755 – 779 (1988).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
334 I. E. WIJAYANTI, M. ARDIYANSYAH, P. W. PRASETYO
9. M. D. Larsen, P. J. McCarthy, Multiplicative theory of ideals, Acad. Press, Inc., USA (1971).
10. S. R. Lopez-Permouth, J. E. Simental, Characterizing rings in terms of the extent of the injectivity and projectivity
of their modules, J. Algebra, 362, 56 – 69 (2012).
11. B. Saraç, P. F. Smith, Y. Tiraş, On Dedekind modules, Commun. Algebra, 35, 1533 – 1538 (2007).
12. P. F. Smith, Mapping between module lattices, Int. Electron. J. Algebra, 15, 173 – 195 (2014).
13. U. Tekir, On multiplication modules, Int. Math. Forum, 2(29), 1415 – 1420 (2007).
14. I. E. Wijayanti, R. Wisbauer, Coprime modules and comodules, Commun. Algebra, 37, № 4, 1308 – 1333 (2009).
15. R. Wisbauer, Grundlagen der Modul- und Ringtheorie: ein Handbuch für Studium und Forschung, Verlag R. Fischer,
München (1988).
16. S. Yassemi, The dual notion of prime submodules, Arch. Math. (Brno), 37, 273 – 278 (2001).
Received 25.02.17
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| last_indexed | 2026-03-24T02:02:57Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/76/4d6973688a470aa1e0dd33b2cdcc2f76.pdf |
| spelling | umjimathkievua-article-5132025-03-31T08:48:21Z On a class of $\lambda$ -modules On a class of $\lambda$ -modules Wijayanti, I. E. Ardiyansyah, M. Prasetyo, P. W. I. E. Prasetyo, P. W. Wijayanti, I. E. Ardiyansyah, M. Prasetyo, P. W. lattice of ideals lattice of submodules multiplication modules class of modules lattice of ideals lattice of submodules multiplication modules class of modules UDC 512.5 Smith in paper [{\it Mapping between module lattices}, Int. Electron. J. Algebra, {\bf 15}, 173–195 (2014)] introduced maps between the lattice of ideals of a commutative ring and the lattice of submodules of an $R$-module $M,$ i.e., $\mu$ and $\lambda$ mappings. The definitions of the maps were motivated by the definition of multiplication modules. Moreover, some sufficient conditions for the maps to be a lattice homomorphisms are studied. In this work we define a class of $\lambda$-modules and observe the properties of the class. We give a sufficient conditions for the module and the ring such that the class $\lambda$ is a hereditary pretorsion class. УДК 512.5 Про один клас $\lambda$ -модулiв У роботі [Mapping between module lattices, Int. Electron. J. Algebra, 15, 173–195 (2014)] Сміт увів у розгляд відображення між решіткою ідеалів комутативного кільця та решіткою субмодулів $R$-модуля $M,$ тобто відображення $\mu$ і $\lambda$. Ці означення були мотивовані означеннями мультиплікативних модулів. Також було вказано деякі достатні умови, за яких ці відображення є гомоморфізмами решіток. У цій роботі наведено означення класу $\lambda$-модулів та зазначено властивості цього класу. Вказано достатні умови на модуль та кільце, за яких клас $\lambda$ є спадковим преторсійним класом.   Institute of Mathematics, NAS of Ukraine 2021-03-11 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/513 10.37863/umzh.v73i3.513 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 3 (2021); 329 - 334 Український математичний журнал; Том 73 № 3 (2021); 329 - 334 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/513/8976 |
| spellingShingle | Wijayanti, I. E. Ardiyansyah, M. Prasetyo, P. W. I. E. Prasetyo, P. W. Wijayanti, I. E. Ardiyansyah, M. Prasetyo, P. W. On a class of $\lambda$ -modules |
| title | On a class of $\lambda$ -modules |
| title_alt | On a class of $\lambda$ -modules |
| title_full | On a class of $\lambda$ -modules |
| title_fullStr | On a class of $\lambda$ -modules |
| title_full_unstemmed | On a class of $\lambda$ -modules |
| title_short | On a class of $\lambda$ -modules |
| title_sort | on a class of $\lambda$ -modules |
| topic_facet | lattice of ideals lattice of submodules multiplication modules class of modules lattice of ideals lattice of submodules multiplication modules class of modules |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/513 |
| work_keys_str_mv | AT wijayantiie onaclassoflambdamodules AT ardiyansyahm onaclassoflambdamodules AT prasetyopw onaclassoflambdamodules AT ie onaclassoflambdamodules AT prasetyopw onaclassoflambdamodules AT wijayantiie onaclassoflambdamodules AT ardiyansyahm onaclassoflambdamodules AT prasetyopw onaclassoflambdamodules |