On some commutative invariants of modules over minimax nilpotent groups
In the paper we introduce a finite system of invariants for modules over minimax nilpotent groups which consists of classes of equivalent prime ideals of the group algebra of an Abelian minimax group. In particuly, introduced system of invariants allows to study the structure of a minimax nilpoten...
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irk-123456789-1871702022-12-09T01:26:34Z On some commutative invariants of modules over minimax nilpotent groups Tushev, A.V. Математика In the paper we introduce a finite system of invariants for modules over minimax nilpotent groups which consists of classes of equivalent prime ideals of the group algebra of an Abelian minimax group. In particuly, introduced system of invariants allows to study the structure of a minimax nilpotent group N acted by a group of operators G such that N has injective modules which are stabilized by the group of operators G. У статті введено скінченну множину інваріантів для модулів над мінімаксними нільпотентними групами, що складається з класів еквівалентних простих ідеалів групової алгебри абелевої мінімаксної групи. Введена множина інваріантів дає змогу, зокрема, вивчати будову мінімаксної нільпотентної групи N, на якій діє група операторів G, причому N має ін’єктивні модулі, які стабілізуються групою операторів G. 2022 Article On some commutative invariants of modules over minimax nilpotent groups / A.V. Tushev // Доповіді Національної академії наук України. — 2022. — № 4. — С. 19-24. — Бібліогр.: 12 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2022.04.019 http://dspace.nbuv.gov.ua/handle/123456789/187170 512.544 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Математика Математика Tushev, A.V. On some commutative invariants of modules over minimax nilpotent groups Доповіді НАН України |
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In the paper we introduce a finite system of invariants for modules over minimax nilpotent groups which consists of
classes of equivalent prime ideals of the group algebra of an Abelian minimax group. In particuly, introduced system
of invariants allows to study the structure of a minimax nilpotent group N acted by a group of operators G such that
N has injective modules which are stabilized by the group of operators G. |
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| author |
Tushev, A.V. |
| author_facet |
Tushev, A.V. |
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Tushev, A.V. |
| title |
On some commutative invariants of modules over minimax nilpotent groups |
| title_short |
On some commutative invariants of modules over minimax nilpotent groups |
| title_full |
On some commutative invariants of modules over minimax nilpotent groups |
| title_fullStr |
On some commutative invariants of modules over minimax nilpotent groups |
| title_full_unstemmed |
On some commutative invariants of modules over minimax nilpotent groups |
| title_sort |
on some commutative invariants of modules over minimax nilpotent groups |
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Видавничий дім "Академперіодика" НАН України |
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2022 |
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Математика |
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| citation_txt |
On some commutative invariants of modules over minimax nilpotent groups / A.V. Tushev // Доповіді Національної академії наук України. — 2022. — № 4. — С. 19-24. — Бібліогр.: 12 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT tushevav onsomecommutativeinvariantsofmodulesoverminimaxnilpotentgroups |
| first_indexed |
2025-07-16T08:36:20Z |
| last_indexed |
2025-07-16T08:36:20Z |
| _version_ |
1837791949716717568 |
| fulltext |
19ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 4: 19—24
Ц и т у в а н н я: Tushev A.V. On some commutative invariants of modules over minimax nilpotent groups. Допов.
Нац. акад. наук Укр. 2022. № 4. С. 19—24. https://doi.org/10.15407/dopovidi2022.04.019
https://doi.org/10.15407/dopovidi2022.04.019
UDK 512.544
A.V. Tushev, https://orcid.org/0000-0001-5219-1986
Oles Honchar Dnipro National University
E-mail: anavlatus@gmail.com
On some commutative invariants
of modules over minimax nilpotent groups
Presented by Corresponding Member of the NAS of Ukraine V.P. Motornyi
In the paper we introduce a finite system of invariants for modules over minimax nilpotent groups which consists of
classes of equivalent prime ideals of the group algebra of an Abelian minimax group. In particuly, introduced system
of invariants allows to study the structure of a minimax nilpotent group N acted by a group of operators G such that
N has injective modules which are stabilized by the group of operators G.
Keywords: nilpotent groups, minimax groups, group rings.
A group G is said to have finite (Prufer) rank if there is a positive integer m such that any finitely
generated subgroup of G may be generated by m elements; the smallest m with this property is
the rank r(G) of G. A group G is said to be of finite torsion-free rank if it has a finite series each of
whose factor is either infinite cyclic or locally finite; the number r0(G) of infinite cyclic factors in
such a series is the torsion-free rank of G. If the group G has a finite series each of whose factor is
either cyclic or quasi-cyclic then G is said to be minimax. If in such a series all factors are cyclic
then the group G is said to be polycyclic.
Let H be a subgroup of a group G, the subgroup H is said to be dense in G if for any g G∈
there is an integer n∈ such that ng H∈ . If \ng G H∈ for any n∈ and any \g G H∈ then
the subgroup H is said to be isolated in G. If the group G is locally nilpotent then the isolator
is ( ) { | for some }n
G H g G g H n= ∈ ∈ ∈ of H in G is a subgroup of G and if H is a normal sub-
group then so is is ( )G H .
We need some notations introduced by Wilson in [1, section 3.8] and based on the results of
[2, 3]. Let A be a torsion-free abelian group of finite rank and let k be a field. If I and J are ideals
of kA then we write I ≈ J if I kB = J kB for some finitely generated dense subgroup B A.
Then ≈ is an equivalence relation on the set of all ideals of kA and we denote by [ ]J the class of
equivalence containing an ideal J of kA. If the ideal J is proper we will say that the class [ ]J is
МАТЕМАТИКА
MATHEMATICS
20 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 4
A.V. Tushev
proper. If a group G acts on A then we obtain an action of G on the set of equivalent prime ideals
of kA which is given by [ ] [ ]g gJ J= .
If B is a dense subgroup of A and P is a prime ideal of kB then, as kA is an integer domain over kB,
it follows from [4, Chap. V, § 2 Theorem 1] that there is a prime ideal Q of kA such that Q kB = P
and we put [ ][ ]kAP Q= . If μ is a set of prime ideals of kB then we put ∈μ = μ[ ] {[ ] }kA kAP P .
Let S be a commutative ring and let J be an ideal of S. Then μS( J) denotes the set of all prime
ideals of S, which are minimal over J. If the ring S is Noetherian then it follows from [4, Chap. IV,
§ 1.4, Theorem 2] that the set μS(J) is finite.
Let J be an ideal of kA. We say that a finitely generated dense subgroup C of A is J-local
if for any subgroup X of finite index in C the mapping ( ) ( )kC kXJ kC J kXμ →μ given by
P P kX is bijective. By [5, Theorem 3.7], the group ring kC is a Noetherian and hence the
set μkC(J kC) is finite.
Proposition 1. Let A be a torsion-free abelian group of finite rank and let k be a field. Let B be
a dense subgroup of A and let J be an ideal of kB. Then:
(i) B contains a J-local subgroup;
(ii) for any J-local subgroup C B we have [ ( ) ] ( )kC kA kCJ kC J kCμ = μ < ∞ ;
(iii) for any J-local subgroups C, D B we have [ ( ) [] ( )]kC kA kD kAJ kC J kDμ = μ .
Let G be a group and let K be a normal subgroup of G. Let R be a ring and let I be a G-invariant
ideal of the group ring RK then † ( 1)I G I= + is a G-invariant subgroup of G.
We say that the ideal I is G-large if R/(R I) = k is a field, †K I < ∞ and I = (RF I)RK,
where F is a G-invariant subgroup of K such that †I F and the quotient group †F I is abelian.
If R is a field then, certainly, R = k.
Let N be a normal subgroups of G such that K N G and the quotient group N/K is tor-
sion-free abelian of finite rank. Let k be a field and let I be a G-large ideal of kK. Then, as the
quotient group †K I is finite, the derived subgroup of the quotient group †N I is finite and it is
not difficult to show that †N I has a characteristic central subgroup A of finite index. Since the
quotient group N/K is torsion-free abelian, the subgroup A may be taken torsion-free. Let W be a
kN-module then W/WI may be considered as kA-module. Since the group algebra kA is commuta-
tive, we can apply methods of commutative algebra for studying the module W. This approach was
introduced by Brookes in [6] for the case where the group G is polycyclic-by-finite.
Let L be a dense subgroups of N such that K L and the quotient groups L/K is finitely ge-
nerated. Then †B A L I= is a dense central finitely generated torsion-free subgroup of A. By
[5, Theorem 3.7], the group ring kB is a Noetherian. Let V be a finitely generated RL-module then
V / VI is a finitely generated kB -module and hence V/VI is a Noetherian kB-module.
By [4, Chap. IV, § 1.4, Theorem 2], for any commutative Noetherian ring S and any Noe-
t herian S-module M the set μS(M) = μS(AnnSM) of prime ideals of S, which are minimal over
AnnSM, is finite. Therefore, we can define a finite set μkB(V/VI) = μkB(AnnkB(V/VI)) of prime
ideals of kB.
Let W be a kN-module which is kK-torsion-free and let akL be a cyclic kL-module generated by
an element 0 ≠ a ∈ W. Let aL be the image of the element a in the quotient module akL/akLI. By
Proposition 1(i), there is a finitely generated dense AnnkA(aL)-local subgroup †
LA A L I B= .
As AL is a finitely generated dense subgroup of finite index in †A L I , we can conclude
that AL is a dense finitely generated torsion-free subgroup of A and akL/akLI is a finitely gene-
21ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 4
On some commutative invariants of modules over minimax nilpotent groups
rated kAL-module. Then it follows from [5, Theorem 3.7] that the domain kAL is Noetherian and
akL/akLI is a Noetherian kAL-module. Thus, the finite set ( / )
LkA akL akLIμ is well defined.
As †/ ( )LakL akLI a k L I= and AL is a central subgroup of finite index in †L I , we see that
Ann ( / ) Ann ( )
L LkA kA LakL aklI a= and hence
( / ) (Ann ( / )) (Ann ( )).
L L L L LkA kA kA kA kA LakL akLI akL aklI aμ = μ = μ (1)
Thus, the set ( / )
LkA akL akLIμ is defined for any 0 ≠ a ∈ W and we put [ ( / )]
LkA kAakL akLIμ =
{[ ] | ( / )}
LkA kAP P akL akLI= ∈μ . It follows from (1) that
[ ( / )] [ (Ann ( / ))] [ (Ann ( ))]
L L L L LkA kA kA kA kA kA kA L kAakL akLI bkL bkLI aμ = μ = μ . (2)
Then, by (2) and according to Proposition 1 (ii), (iii), the set [ ( / )]
LkA kAakL akLIμ =
{[ ] | ( / )}
LkA kAP P akL akLI= ∈μ is finite and does not depend on the choice of the AnnkA(aL)-
local subgroup AL. Note that everywhere below in the definition of the set [ ( / )]kB kAakL akLIμ =
{[ ] | ( / )} {[ ] | (Ann ( ))}
L L LkA kA kA kA kA LP P akL akLI P P a= ∈μ = ∈μ we assume that the subgroup
AL is AnnkA(aL)-local.
Proposition 2. Let N be a minimax torsion-free nilpotent group and let K be a normal subgroup
of N such that the quotient group N/K is torsion-free abelian. Let k be a field of characteristic zero
and let W be a kN-module which is kK-torsion-free. Let I be an N-large ideal of kK and A be a
torsion-free characteristic central subgroup of finite index in †N I . Then there exists a cyclic kN-
submodule 0 ≠ V W such that [ ( / )] [ ( / )]
L MkA kA kA kAakL akLI bkM bkMIμ = μ for any elements
0 ≠ a,b V and any dense subgroups L,M N such that K L ∩ M and the quotient groups L/K
and M/K are finitely generated.
The above Proposition shows that we can define the set ( / ) [ ( / )]
LkA kA kAM V V I akL akLI= μ
which does not depends on the choice of an element 0 ≠ a ∈ V and a dense subgroup L N
such that K L and the quotient groups L/K is finitely generated. So, the set MkA(V/VI)
may be considered as a finite set of invariants of the module V. By the definition, the set
( / ) [ ( / )]
LkA kA kAM V V I akM akMI= μ consists of equivalence classes which are defined by prime
ideals of the commutative group algebra kA. The following Theorem considers some important
properties of the set MkA(V/VI).
Theorem 1. Let G be a soluble group of finite torsion-free rank, let N be a minimax nilpotent
torsion-free normal subgroup of G and let K be a G-invariant subgroup of N such that the quotient
group N/K is torsion-free abelian. Let k be a field of characteristic zero and let W be a kN-module
which is kN-torsion and kK-torsion-free. Let X be an isolated subgroup of N such that K X and the
module W is kX-torsion-free. Then there are a cyclic kN-submodule 0 ≠ V W, a G-large ideal I of
kK and a central G-invariant subgroup A of finite index in †/N I such that:
(i) MkA(V/VI) = MkA(akN/akNI) for any element 0 ≠ a ∈ V;
(ii) the kA-module V/VI is kA-torsion but not kB-torsion, where †( / )B A X I= ;
(iii) for any g ∈ G we have ( ) ( ) {[ ] [ ] ( )}/ / [ ] /g g g
kA kA kAM V g V gI M V V I P P P M V V I= = ∈= .
Let A be an abelian torsion-free group acted by a group G, we consider A as an additive
group. Let A A= ⊗ , we denote by Soc ( )G A the socle of the G -module A and put
22 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 4
A.V. Tushev
Soc ( ) Soc ( )G GA A A= . It is not difficult to show that SocG(A) is an isolated G-invariant sub-
group of A.
Let k be a field and let I be an ideal of kA. A subgroup SG(I) G which consists of all ele-
ments g ∈ G such that I kB = Ig kB for some finitely generated dense subgroup B A is
said to be the standartizer of I in G (see [4]). Since the class [ ]I consists of all ideals J kA
such that that I kB = J kB for some finitely generated dense subgroup B A, we see that
[ ] { [ ]}g gI J J I= ∈ also forms the class [ ]gI . So, we have an action of G on the set of equivalence
classes of ideals of kA. It immediately follows from the definition of SG(I) that )( ) ([ ]G GS I S I= ,
where G ([ ]) { [ ] [ ]}gS I G I I= γ∈ = is the stabilizer of [ ]I in G.
Proposition 3. Let A be an abelian torsion-free group of finite rank acted by a soluble-by-
finite group G, let C = SocGA, let k be a field of characteristic zero and let J be an ideal of kA. Let
P1, P2, …, Pn be ideals of kA such that ( ) { |1 }kB iJ kB P kB i nμ = is the set of minimal prime
ideals over J kB for some finitely generated dense subgroup B of A. If : ([ ])G iG S P < ∞ for all
1 i n then J k(C B) ≠ 0.
Let S be a ring and let M, X and Y be S-modules. The modules X and Y are separated in M if X
and Y don’t have nonzero isomorphic sections which are isomorphic to a submodule of M. A sub-
module U of M is said to be essential if U V ≠ 0 for any nonzero submodule V of M. The module
M is said to be uniform if anynonzero submodule of M is essential.
We say that a submodule X M is solid in M if X is uniform and M does not have submodules
which are isomorphic to a proper section of X.
Let N be a normal subgroup of a group G and let R be a ring. Let M and W be RN-modules. A
subgroup Sep(G,N)(M,W) G generated by all elements g ∈ G such that RN-modules W and Wg
are not separated in M is said to be the separator of W in G. Evidently, for any element h ∈ G such
that ( , )Sep ( , )G Nh M W∉ modules W and W h-are separated in M (see [7, 8]).
RN-modules W and V are said to be similar if their injective hulls [W] and [V] are isomorphic.
The modules W and V are similar if and only if they have isomorphic essential submodules. By
Lemma 3.2 of [9], the stabilizer Stab [ ] { | and are similar}G W g G W g W= ∈ of W in G is a sub-
group of G. It is easy to note that StabG[W] Sep(G,N)(M,W), moreover, if the submodule W is
solid in M then StabG[W] = Sep(G,N)(M,W).
Suppose that the module W and the group G satisfies the conditions of Theorem2 and
StabG[W] = G. Suppose also that the module W is uniform. Let I be a G-large ideal of kK and
A be a torsion-free characteristic central subgroup of finite index in †N I . By Theorem1,
there are a cyclic kN-submodule 0 ≠ V W, a G-large ideal I of kK and a central G-invariant
subgroup A of finite index in †N I such that MkA(V/VI) = MkA(akN/kNI) for any 0 ≠ a ∈ V
and MkA(Vg/VgI) = MkA(V/VI)g for any g ∈ G. As the module W is uniform, it is not difficult
to note that StabG[V] = G and hence there are nonzero cyclic submodules akN and bkN of V
such that akN ≅ (bkN)g. Then it follows from Theorem 1 that MkA(V/VI) = MkA(akN/akNI) =
= MkA((bkN)g/(bkN)gI) = MkA(bkN/bkNI)g = MkA(V/VI)g and hence MkA(V/VI) = MkA(V/VI)g
for any g ∈ G. So, we have an action of the group G on the finite set MkA(V/VI). By the
definition of MkA(V/VI), there are prime ideals P1, P2, …, Pn of kA such that ( / )kAM V V I =
{[ ] 1 }iP i n= . Therefore, as MkA(V/VI) = MkA(V/VI)g for any g ∈ G, we can conclude that
: ([ ])G iG S P < ∞ for all 1 i n. Then we can apply Proposition 3 which plays a central role
in the proof of the following Theorem.
23ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 4
On some commutative invariants of modules over minimax nilpotent groups
Theorem 2. Let N be a nilpotent normal non-abelian minimax torsion-free subgroup of a solv-
able-by-finite group G of finite torsion-free rank. Let K be a G-invariant subgroup of N such that the
quotient group N/K istorsion-free abelian. Let k be a field of characteristic zero. Suppose that there is
a uniform kN-torsion kN-module W such that StabG[W] = G and for any proper isolated G-invariant
subgroup X of N such that K X the module W is kX-torsion-free. Then SocG(N/K) = N/K.
Let R be a ring and let M be an R-module then KR(M) denotes the Krull dimension of M. The
module M is said to be ρ-critical if KR(M) = ρ and KR(M/U) < KR(M) = ρ for any non-zero sub-
module U M.
Let N be a nilpotent group of finite torsion-free rank, let k be a field and let M be an kN-mo-
dule. Let S be a finitely generated subring of k. Let 0 ≠ a ∈ M and let H be a proper isolated normal
subgroup of N. We say that (a, H) is an important pair for the SN-module M if there is a finitely
generated dense subgroup A N such that:
(i) the module aSA is ρ -critical and ( ) ( )SX SXK aSA K xSX for any element 0 x M≠ ∈
and any finitely generated dense subgroup X N ;
(ii) ,SBaSA aSB SA= ⊗ where B = A H;
(iii) if 0 ≠ b ∈ aSB and V is a dense subgroup of B then there is no isolated subgroup D V
such that SDbSV bSD SV= ⊗ and isN(D) is a normal subgroup of N.
The module M is said to be impervious if it has no important pairs for any finitely generated
subring S of k.
Theorem 3. Let N be a nilpotent non-abelian minimax torsion-free normal subgroup of a sol-
uble-by-finite group G of finite torsion-free rank. Let K be a G-invariant subgroup of N such that
the quotient group N/K is torsion-free abelian. Let k be a field of characteristic zero. Let W be a
kG-module which is kN-torsion and for any proper isolated G-invariant subgroup X of N such that
K X the module W is kX-torsion-free. Suppose that W is impervious as a kN-module. If Sep(G,Y)
(xkG, xkY) = G for any element 0 ≠ x ∊ W and any G-invariant subgroup Y of N then:
(i) SocG(N/K) = N/K;
(ii) there is a dense G-invariant subgroup D N such that K D and the quotient group D/K
is polycyclic.
Theorem 4. Let N be a nilpotent non-abelian minimax torsion-free normal subgroup of a sol-
uble-by-finite group G of finite torsion-free rank. Let k be a field of characteristic zero. Let W be a
kG-module which is kN-torsion and for any proper isolated G-invariant subgroup X of N the module
W is kX-torsion-free. Suppose that W is impervious as a kN-module and Sep(G,Y)(xkG, xkY) = G for
any element 0 ≠ x ∈ W and any G-invariant subgroup Y of N. Then for any finitely generated sub-
group H of G the subgroup N has an H-invariant polycyclic dense subgroup.
Corollary 1. Let N be a nilpotent non-abelian minimax torsion-free normal subgroup of a fi-
nitely generated linear group G of finite rank. Let k be a field of characteristic zero, let W be a kG-
module which is kN-torsion and for any proper isolated G-invariant subgroup X of N the module W is
kX-torsion-free. Suppose that W is impervious as a kN-module and Sep(G,Y)(xkG, xkY) = G for any
element 0 ≠ x ∈ W and any G-invariant subgroup Y of N. Then there are a G-invariant polycyclic
dense subgroup H of N and an element 0 ≠ a ∈ W such that:
(i) kHakN akH kN= ⊗ is a uniform kN-module;
(ii) the kH-module akH is uniform, impervious and StabG[akH] = G.
Impervious modules over group rings of polycyclic groups were considered in [9-12].
24 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2022. № 4
A.V. Tushev
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Received 22.02.2022
А.В. Тушев, https://orcid.org/0000-0001-5219-1986
Дніпровський національний університет ім. Олеся Гончара
E-mail: anavlatus@gmail.com
ПРО ДЕЯКІ КОМУТАТИВНІ ІНВАРІАНТИ МОДУЛІВ
НАД МІНІМАКСНИМИ НІЛЬПОТЕНТНИМИ ГРУПАМИ
У статті введено скінченну множину інваріантів для модулів над мінімаксними нільпотентними групами,
що складається з класів еквівалентних простих ідеалів групової алгебри абелевої мінімаксної групи. Вве-
дена множина інваріантів дає змогу, зокрема, вивчати будову мінімаксної нільпотентної групи N, на якій
діє група операторів G, причому N має ін’єктивні модулі, які стабілізуються групою операторів G.
Kлючові слова: нільпотентні групи, мінімаксні групи, групові кільця.
https://doi.org/10.1016/0022-4049(88)90129-6
https://doi.org/10.1016/S0021-8693(02)00072-8
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