A note on $S$-Nakayama’s lemma
We propose an $S$-version of Nakayama's lemma. Let $R$ be a commutative ring, $S$ a multiplicative subset of $R,$ and $M$ be an $S$-finite $R$-module. Also let $I$ be an ideal of $R.$ We show that if there exists $t\in S$ such that $tM\subseteq IM,$ then$(t^\prime+a)M=0$ for some $t...
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| author | Hamed , A. Hamed , A. Hamed , A. |
| author_facet | Hamed , A. Hamed , A. Hamed , A. |
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| description | We propose an $S$-version of Nakayama's lemma. Let $R$ be a commutative ring, $S$ a multiplicative subset of $R,$ and $M$ be an $S$-finite $R$-module. Also let $I$ be an ideal of $R.$ We show that if there exists $t\in S$ such that $tM\subseteq IM,$ then$(t^\prime+a)M=0$ for some $t'\in S$ and $a\in I.$ We also give an analog of Nakayama's lemma for a $w$-ideal and an $S$-$w$-finite $R$-module, where $R$ is an integral domain. Thus, we generalize the result obtained by Wang and McCasland [Commun. Algebra, 25, 1285–1306 (1997)]. |
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UDC 512.5
A. Hamed (Univ. Monastir, Tunisia)
A NOTE ON \bfitS -NAKAYAMA’S LEMMA
ЗАУВАЖЕННЯ ЩОДО \bfitS -ЛЕМИ НАКАЯМИ
We propose an S -version of Nakayama’s lemma. Let R be a commutative ring, S a multiplicative subset of R, and M
be an S -finite R-module. Also let I be an ideal of R. We show that if there exists t \in S such that tM \subseteq IM, then
(t\prime +a)M = 0 for some t\prime \in S and a \in I. We also give an analog of Nakayama’s lemma for a w-ideal and an S -w-finite
R-module, where R is an integral domain. Thus, we generalize the result obtained by Wang and McCasland [Commun.
Algebra, 25, 1285 – 1306 (1997)].
Запропоновано S -версiю леми Накаями. Нехай R — комутативне кiльце, S — мультиплiкативна пiдмножина R, а
M — S -скiнченний R-модуль. Крiм того, нехай I — iдеал в R. Доведено, що у випадку, коли iснує t \in S таке,
що tM \subseteq IM, маємо (t\prime + a)M = 0 для деяких t\prime \in S та a \in I. Також наведено аналог леми Накаями для
w-iдеалу та S -w-скiнченного R-модуля, де R є iнтегральною множиною. Таким чином, узагальнено результат, що
був отриманий Вангом та МакКасландом [Commun. Algebra, 25, 1285 – 1306 (1997)].
1. Introduction. Let R be a commutative ring with identity and let M be a unitary R-module.
Nakayama’s lemma is a well-known result, which states that every finitely generated R-module M
such that M = IM for some ideal I of R, implies that there exists an a \in I such that (1+a)M = 0
[6] (Theorem 2.2). Some generalizations of Nakayama’s lemma has been given and studied, in the
literatures [1, 3].
In [2], Anderson and Dumitrescu generalized the notion of finitely generated module by intro-
ducing the concept of S -finite modules. Let R be a commutative ring with identity, S \subseteq R be a
given multiplicative set and M be an R-module. We say that M is S -finite if sM \subseteq F for some
finitely generated submodule F of M and some s \in S. Note that if S consists of units of R, then
M is an S -finite module if and only if M is a finitely generated module.
The first result of this paper is to try to relaxing the condition finitely generated for M with
weaker condition (S -finite) and we will study the Nakayama’s lemma. Let R be a commutative ring,
S be a multiplicative subset of R and M be an S -finite R-module. Let I be an ideal of R. We show
that if there exist t \in S such that tM \subseteq IM, then (t\prime + a)M = 0 for some t\prime \in S and a \in I. Also,
if there exist t \in S such that tM \subseteq IM +N, for some submodule N of M, then (t\prime + a)M \subseteq N
for some t\prime \in S and a \in I. Note that if S consists of units of R we find the Nakayama’s lemma.
On the other hand, in [4], the authors gived an analogue of the Nakayama’s lemma for a w-ideal
and a w-module of finite type. First, let us recall the following notions. Let D be an integral
domain with quotient field K and let J be an ideal of D. We say that J is a Glaz – Vasconcelos
ideal (GV-ideal) if J is finitely generated and J - 1 = D. Let GV(D) be the set of GV-ideals of
D. Following [4], a torsion-free D-module M is called a w-module if xJ \subseteq M for J \in GV(D)
and x \in M \otimes K imply that x \in M. M is a w-ideal if M is an ideal of D and is also a w-
module. For a torsion-free D-module M, Wang and McCasland defined the w-envelope of M in
[4] as Mw = \{ x \in M \otimes K | xJ \subseteq M for some J \in GV (D)\} . In particular, if I is a nonzero
fractional ideal of D, then Iw = \{ x \in K | xJ \subseteq I for some J \in GV (D)\} . We say that a torsion-free
R-module M is w-finite type if M = Nw, for some finitely generated submodule N of M. For
a w-finite type R-module M and a proper w-ideal I of D, Wang and McCasland showed that if
c\bigcirc A. HAMED, 2020
142 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
A NOTE ON S -NAKAYAMA’S LEMMA 143
M = (IM)w, then M = 0. In [5], the authors generalized the notion of w-finite type module by
introducing the concept of S -w-finite modules. Let D be an integral domain, S be a multiplicative
subset of D and M be a (torsion-free) w-module as a D-module. We say that M is S -w-finite if
sM \subseteq Fw for some s \in S and some finitely generated submodule F of M. Note that if S consists
of units of D, then M is an S -w-finite module if and only if M is a w-finite module.
The second result of this work gives an analogue of the Nakayama’s lemma for a w-ideal and
an S -w-finite module. Indeed, let D be an integral domain, S be a multiplicative subset of D
and M be an S -w-finite D-module. Let I be a w-ideal of A disjoint with S, we show that, if
tM \subseteq (IM)w for some t \in S, then M = 0. So we generalize the result of Wang and McCasland
[4] (Corollary 2.10).
2. Main results. Let R be a commutative ring, S be a multiplicative subset of R and M be
an R-module. Recall from [2] that M is called S -finite if sM \subseteq F for some finitely generated
submodule F of M and some s \in S. The next result give an S -version of Nakayama’s lemma. Let
M be an S -finite R-module, I be an ideal of R and N be a submodule of M. We show that if
there exist t \in S such that tM \subseteq IM +N, then (t\prime + a)M \subseteq N for some t\prime \in S and a \in I. The
demonstration of this general statement reduces to that of the particular case N = 0.
Lemma 2.1. Let R be a commutative ring, S be a multiplicative subset of R and M be an S -
finite R-module. Let I be an ideal of R. If there exist t \in S such that tM \subseteq IM, then (t\prime +a)M = 0
for some t\prime \in S and a \in I.
Proof. As M is S -finite, then there exist an s \in S and a finitely generated submodule F =
= a1R + . . . + anR of M such that sM \subseteq F \subseteq M. We have stM \subseteq sIM \subseteq IF. Then, for all
1 \leq i \leq n, stai =
\sum n
j=1
yi,jaj with yi,j \in I. Noting Y the matrix of yi,j and d the determinant
of stIn - Y. By Laplace formula we have, for all 1 \leq i \leq n, dai = 0, hence d(sM) = 0,
because sM \subseteq F. Moreover, by developing the determinant it’s easy to see that d \in (st)n + I, then
sd \in sn+1tn + I, thus sd = t\prime + a where t\prime = sn+1tn \in S and a \in I.
If S included in the set of units of D, we find the following corollary, which is a particular case
of Nakayama’s lemma (N = 0) [6].
Corollary 2.1. Let R be a commutative ring, I be an ideal of R and M be a finitely generated
module over R. If M = IM, then (1 + a)M = 0 for some a \in I.
Our next result give an S -version of Nakayama’s lemma.
Theorem 2.1. Let R be a commutative ring, S be a multiplicative subset of R and M be an
S -finite R-module. Let I be an ideal of R and N be a submodule of M. If there exist t \in S such
that tM \subseteq IM +N, then (t\prime + a)M \subseteq N for some t\prime \in S and a \in I.
Proof. We put N \prime = M\diagup N. Since M is an S -finite R-module, then N \prime is also an S -finite
R-module. Moreover, tN \prime = tM\diagup N \subseteq (IM +N)\diagup N \subseteq IM\diagup N = IN \prime . Thus by Lemma 2.1,
there exist a t\prime \in S and an a \in I such that (t\prime + a)N \prime = (t\prime + a)M\diagup N = 0, which implies that
(t\prime + a)M \subseteq N.
Corollary 2.2. Let M be a finitely generated module over R, I be an ideal of R and N be a
submodule of M such that M \subseteq IM +N, then (1 + a)M \subseteq N for some a \in I.
Remark 2.1. Let R be a commutative ring, S be a multiplicative subset of R and M be an
S -finite R-module. Let N be a submodule of M and I be an ideal of R contained in the Jacobson
radical of R. If there exist t \in S such that tM \subseteq IM+N, then by the previous theorem, (t\prime +a)M \subseteq
\subseteq N for some t\prime \in S and a \in I.
Remark that if S consists of units of R, then t\prime + a is an unit of R, thus M = N.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
144 A. HAMED
The next two results are analogues of the S -version of Nakayama’s lemma for a w-ideal and an
S -w-finite D-module.
Theorem 2.2. Let D be an integral domain, S be a multiplicative subset of D and M be an
S -finite torsion-free D-module. Let I be a w-ideal of D disjoint with S. If tMw \subseteq (IM)w for some
t \in S, then M = 0.
Proof. As M is S -finite, then there exists an s \in S and a finitely generated submodule F =
= a1D+ . . .+anD of M such that sM \subseteq F \subseteq M. Since F is a finitely generated and tF \subseteq tM \subseteq
\subseteq tMw \subseteq (IM)w, then tJF \subseteq IM for some J \in GV(D), which implies that stJM \subseteq IM. Let
r \in J. For all 1 \leq i \leq n, we have s2trai \in s2tJM \subseteq sIM \subseteq IF, then, for all 1 \leq i \leq n, s2trai =
=
\sum n
j=1
yi,jaj with yi,j \in I. Noting Y the matrix of yi,j and d the determinant of s2trIn - Y.
By Laplace formula, we have, for all 1 \leq i \leq n, dai = 0, hence d(sM) = 0, because sM \subseteq F.
By developing the determinant it’s easy to see that d \in (s2tr)n + I, then sd \in s2n+1tnrn + I. If
M \not = 0, then sd = 0 (since sdM = 0) which implies that t\prime rn \in I where t\prime = s2n+1tn \in S. Thus,
t\prime J \subseteq
\surd
I, because for each r \in J, we have (t\prime r)n = (t\prime )n - 1t\prime rn \in I. Therefore, by [4] (Proposition
1.6), t\prime \in (
\surd
I)w =
\surd
I, then there exists a q \in \BbbN , such that (t\prime )q \in I hence (t\prime )q \in S \cap I, a
contradiction.
Let D be an integral domain, S be a multiplicative subset of D and M be a (torsion-free)
w-module as D-module. We say that M is S -w-finite if sM \subseteq Fw for some s \in S and some
finitely generated submodule F of M ([5]).
Corollary 2.3. Let D be an integral domain, S be a multiplicative subset of D and M be an
S -w-finite D-module. Let I be a w-ideal of A disjoint with S. If tM \subseteq (IM)w for some t \in S,
then M = 0.
Proof. Since M is S -w-finite, then there exist an s \in S and a finitely generated submodule F of
M such that sM \subseteq Fw \subseteq M. We have stFw \subseteq stM \subseteq s(IM)w = (sIM)w \subseteq (IFw)w = (IF )w
[4] (Proposition 2.8). Then stFw \subseteq (IF )w. Since F is a finitely generated submodule of M, then
F is an S -finite D-module. Moreover, F is a torsion-free D-module. Thus by the previous theorem
F = 0, which implies that sM = 0, hence M = 0.
In the particular case when S consists of units of D, we find the result of Wang and McCasland
[4] (Corollary 2.10).
Corollary 2.4. Let D be an integral domain, M be a w-finite type D-module and I be a proper
w-ideal of D. If M = (IM)w, then M = 0.
References
1. R. Ameri, Two versions of nakayama lemma for multiplication modules, Int. J. Math. Math. Sci., 54, 2911 – 2913
(2004).
2. D. D. Anderson, T. Dumitrescu, S -Noetherian rings, Commun. Algebra, 30, 4407 – 4416 (2002).
3. A. Azizi, On generalization of Nakayama’s lemma, Glasg. Math. J., 52, 605 – 617 (2010).
4. W. Fanggui, R. L. McCasland, On w-modules over strong Mori domains, Commun. Algebra, 25, 1285 – 1306 (1997).
5. H. Kim, M. O. Kim, J. W. Lim, On S -strong Mori domains, J. Algebra, 416, 314 – 332 (2014).
6. H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, UK (1992).
Received 18.07.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
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| id | umjimathkievua-article-2332 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:18Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-23322020-01-27T10:49:35Z A note on $S$-Nakayama’s lemma Зауваження щодо $S$-леми Накаями Зауваження щодо $S$ -леми Накаями Hamed , A. Hamed , A. Hamed , A. Nakayama’s lemma S-finite S-w-finite We propose an $S$-version of Nakayama's lemma. Let $R$ be a commutative ring, $S$ a multiplicative subset of $R,$ and $M$ be an $S$-finite $R$-module. Also let $I$ be an ideal of $R.$ We show that if there exists $t\in S$ such that $tM\subseteq IM,$ then$(t^\prime+a)M=0$ for some $t'\in S$ and $a\in I.$ We also give an analog of Nakayama's lemma for a $w$-ideal and an $S$-$w$-finite $R$-module, where $R$ is an integral domain. Thus, we generalize the result obtained by Wang and McCasland [Commun. Algebra, 25, 1285–1306 (1997)]. Запропоновано $S$-версію леми Накаями. Нехай $R$ – комутативне кільце, $S$ – мультиплікативна підмножина $R,$ а $M$ – $S$-скінченний $R$-модуль. Крім того, нехай $I$ – ідеал в $R.$ Доведено, що у випадку, коли існує $t\in S$ таке, що $tM\subseteq IM,$ маємо $(t'+a)M=0$ для деяких $t'\in S$ та $a\in I.$ Також наведено аналог леми Накаями для $w$-ідеалу та $S$-$w$-скінченного $R$-модуля, де $R$ є інтегральною множиною. Таким чином, узагальнено результат, що був отриманий Вангом та МакКасландом [Commun. Algebra, 25, 1285–1306 (1997)]. Запропоновано $S$-версію леми Накаями. Нехай $R$ – комутативне кільце, $S$ – мультиплікативна підмножина $R,$ а $M$ – $S$-скінченний $R$-модуль. Крім того, нехай $I$ – ідеал в $R.$ Доведено, що у випадку, коли існує $t\in S$ таке, що $tM\subseteq IM,$ маємо $(t'+a)M=0$ для деяких $t'\in S$ та $a\in I.$ Також наведено аналог леми Накаями для $w$-ідеалу та $S$-$w$-скінченного $R$-модуля, де $R$ є інтегральною множиною. Таким чином, узагальнено результат, що був отриманий Вангом та МакКасландом [Commun. Algebra, 25, 1285–1306 (1997)]. Institute of Mathematics, NAS of Ukraine 2020-01-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2332 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 1 (2020); 142-144 Український математичний журнал; Том 72 № 1 (2020); 142-144 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2332/1553 Copyright (c) 2020 Elina Dichter (Менеджер журналу) |
| spellingShingle | Hamed , A. Hamed , A. Hamed , A. A note on $S$-Nakayama’s lemma |
| title | A note on $S$-Nakayama’s lemma |
| title_alt | Зауваження щодо $S$-леми Накаями Зауваження щодо $S$ -леми Накаями |
| title_full | A note on $S$-Nakayama’s lemma |
| title_fullStr | A note on $S$-Nakayama’s lemma |
| title_full_unstemmed | A note on $S$-Nakayama’s lemma |
| title_short | A note on $S$-Nakayama’s lemma |
| title_sort | note on $s$-nakayama’s lemma |
| topic_facet | Nakayama’s lemma S-finite S-w-finite |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2332 |
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