Groups associated with modules over nearrings
We construct a group D(I, T) associated with the pair (I, T), where I is a nontrivial distributive submodule of a left N-module G, T is a nontrivial subgroup of the unit group U(N) of a right nearring N with an identity element, and find criteria for D(I, T) to be a Frobenius group. We construc...
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Інститут прикладної математики і механіки НАН України
2007
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | Groups associated with modules over nearrings / O.D. Artemovych, I.V. Kravets // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 2. — С. 16–21. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1573412019-06-21T01:30:14Z Groups associated with modules over nearrings Artemovych, O.D. Kravets, I.V. We construct a group D(I, T) associated with the pair (I, T), where I is a nontrivial distributive submodule of a left N-module G, T is a nontrivial subgroup of the unit group U(N) of a right nearring N with an identity element, and find criteria for D(I, T) to be a Frobenius group. We construct a group D(I, T) associated with the pair (I, T), where I is a nontrivial distributive submodule of a left N-module G, T is a nontrivial subgroup of the unit group U(N) of a right nearring N with an identity element, and find criteria for D(I, T) to be a Frobenius group. 2007 Article Groups associated with modules over nearrings / O.D. Artemovych, I.V. Kravets // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 2. — С. 16–21. — Бібліогр.: 4 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/157341 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
We construct a group D(I, T) associated with the
pair (I, T), where I is a nontrivial distributive submodule of a left
N-module G, T is a nontrivial subgroup of the unit group U(N) of
a right nearring N with an identity element, and find criteria for
D(I, T) to be a Frobenius group.
We construct a group D(I, T) associated with the
pair (I, T), where I is a nontrivial distributive submodule of a left
N-module G, T is a nontrivial subgroup of the unit group U(N) of
a right nearring N with an identity element, and find criteria for
D(I, T) to be a Frobenius group. |
| format |
Article |
| author |
Artemovych, O.D. Kravets, I.V. |
| spellingShingle |
Artemovych, O.D. Kravets, I.V. Groups associated with modules over nearrings Algebra and Discrete Mathematics |
| author_facet |
Artemovych, O.D. Kravets, I.V. |
| author_sort |
Artemovych, O.D. |
| title |
Groups associated with modules over nearrings |
| title_short |
Groups associated with modules over nearrings |
| title_full |
Groups associated with modules over nearrings |
| title_fullStr |
Groups associated with modules over nearrings |
| title_full_unstemmed |
Groups associated with modules over nearrings |
| title_sort |
groups associated with modules over nearrings |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157341 |
| citation_txt |
Groups associated with modules over nearrings / O.D. Artemovych, I.V. Kravets // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 2. — С. 16–21. — Бібліогр.: 4 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT artemovychod groupsassociatedwithmodulesovernearrings AT kravetsiv groupsassociatedwithmodulesovernearrings |
| first_indexed |
2025-07-14T09:47:11Z |
| last_indexed |
2025-07-14T09:47:11Z |
| _version_ |
1837615212873646080 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2007). pp. 16 – 21
c© Journal “Algebra and Discrete Mathematics”
Groups associated with modules over nearrings
O. D. Artemovych, I. V. Kravets
Dedicated to V.I. Sushchansky on the occasion of his 60th birthday
Abstract. We construct a group D(I, T ) associated with the
pair (I, T ), where I is a nontrivial distributive submodule of a left
N -module G, T is a nontrivial subgroup of the unit group U(N) of
a right nearring N with an identity element, and find criteria for
D(I, T ) to be a Frobenius group.
0. Let N be a right nearring under two operations “+” and “·” with the
identity element 1, i.e. (N, +) is a group with the zero 0, multiplication
“·” is associative and (y + z) · x = y · x + z · x for all x, y, z ∈ N. As
usual, an additive group (G, +) with the zero e is called a left N -module
if (x + y)g = xg + yg and x(yg) = (xy)g for any g ∈ G and x, y ∈ N . A
subgroup H of G is called an N -submodule (or an N -subgroup) of G if
HN ⊆ H. Recall that an N -module G is abelian if the additive group
(G, +) is abelian and x(g + h) = xg + xh for all g, h ∈ G, x ∈ N. A
submodule I of an N -module G will be called distributive with respect
to subset T of N if x(g + h) = xg + xh for all g, h ∈ G and x ∈ T .
Moreover, G is unitary if 1g = g for any g ∈ G.
As it is well known U(N) = {d ∈ N | d is invertible in N} is a group
under “·” (which is called the unit group of N).
In this paper we construct a group D(I, T ), where I is a nontrivial
distributive submodule of a left N -module G, T is a nontrivial subgroup
of the unit group U(N) of a right nearring N with an identity element,
and find criteria for D(I, T ) to be a Frobenius group.
Throughout this paper, all nearrings are right with an identity element
and all modules are left. If H is a group, F its subgroup and x, y ∈ H,
then [x, y] = x−1y−1xy is the commutator of x and y, yx = x−1yx and
F x = x−1Fx = {yx | y ∈ F}.
Other general notations and conventions in this paper follow closely
those used in [1] and [2].
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.O. D. Artemovych, I. V. Kravets 17
1. Let N be a right nearring and G be a left N -group. If T is a subgroup
of U(N) and I is an N -subgroup of G, then on the set of pairs
D(I, T ) = {(a, b) | a ∈ I, b ∈ T}
we define the algebraic operation by the rule
(a, b)(u, v) = (bu + a, bv). (1)
Lemma 1. Let N be a right nearring with the identity element 1, G an
unitary left N -module with the zero e. If T is a subgroup of the unit
group U(N) of N , I is an N -subgroup which is distributive with respect
to T of G (in particular, I is an abelian N -submodule of G), then D(I, T )
is a group with the identity element (e, 1) under the operation given by
the rule (1) and, moreover,
D(I, T ) = E ⋊ F,
where a subgroup E = {(a, 1) | a ∈ I} is isomorphic to the additive group
I+ of I and F = {(0, b) | b ∈ T} is isomorphic to T .
Proof. The proof is immediate. We remark only that (a, b)−1 = (−b−1a, b−1)
for any a ∈ I and b ∈ T .
Remember that a semidirect product H = E ⋊ F of groups E and F
is called a Frobenius group with a kernel E and a complement F if
F ∩ F g = 1
for all g ∈ H\F and
E\{1} = H\
⋃
h∈H
F h.
The following result extends Theorem 2.3 from [3].
Theorem 2. Let N be a right nearring with the identity element 1 and
the zero 0, G an unitary left N -module with the zero e, T a nontrivial
subgroup of the unit group U(N), I is a nontrivial N -subgroup of G
which is distributive with respect to T . Then
H = D(I, T ) = E ⋊ F
is a Frobenius group with a kernel E and a complement F , where E is
isomorphic to the additive group I+ of I and F is isomorphic to T , if and
only if the following conditions hold:
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.18 Groups associated with modules over nearrings
1. ann(T−1)(i) = {t − 1 ∈ (T − 1) | (t − 1)i = e} = {0} for any
nontrivial element i ∈ I;
2. I = (b − 1)I for every nontrivial element b of T .
Proof. (⇒) Let H = E ⋊ F be a Frobenius group with a kernel E ∼= I+
and a complement F ∼= T. By Lemma 1.1 of [3] for every element a ∈ I
and every element t ∈ T there exists a1 ∈ I such that
(a, 1) = [(a1, 1), (e, t)].
But then
(a, 1) = (−a1, 1)(e, t−1)(a1, 1)(e, t) = (1e − a1, 1t−1)(1e + a1, 1t) =
(−a1, t
−1)(a1, t) = (t−1a1 − a1, t
−1t) = (t−1a1 − a1, 1).
This means that
a = t−1a1 − a1 = (t−1 − 1)a1 ∈ (t−1 − 1)I.
As a consequence I = (t−1 − 1)I for each nontrivial element t ∈ T.
Let a be any nontrivial element of I. Suppose that (b − 1)a = e for
some element b ∈ T. Then
(e, b) = ((b − 1)a, b) = (ba − a, b) = (−a, b)(a, 1) = (1e − a, 1b)(a, 1) =
= (−a, 1)(e, b)(a, 1) = (a, 1)−1(e, b)(a, 1) ∈ F (a,1)
⋂
F.
Since
F (u,v) ∩ F = 〈(e, 1)〉
for each (u, v) ∈ H \ F , we conclude that b − 1 = 0.
(⇐) Suppose that the conditions (1) and (2) are true for nontrivial
elements b ∈ T and a ∈ I. Since
a = (b − 1)a1
for some elements a1 ∈ I, we deduce that
[(a1, 1), (e, b−1)] = (−a1, 1)(−be, b)(a1, 1)(e, b−1) =
(−a1, 1)(e, b)(a1, 1)(e, b−1) = (1e − a1, 1 · b)(1e + a1, 1b−1) =
(−a1, b)(a1, b
−1) = (ba1 − a1, bb
−1) = (a, 1).
This yields that E = [E, (e, b)] for any nontrivial element (e, b) ∈ F .
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.O. D. Artemovych, I. V. Kravets 19
Let x ∈ I and t, y ∈ T , where t 6= 1. Then
(e, t)(x,y) = ((y−1t − y−1)x, y−1ty) /∈ E.
Suppose that (c, t) ∈ H \
⋃
(u,v)∈H F (u,v). Inasmuch as (t − 1)I = I,
there exists an element x ∈ I such that c = (t − 1)y−1x and so (c, t) =
(e, yty−1)(x,y). Hence
E \ {(e, 1)} = H \
⋃
(x,y)∈H
F (x,y).
Now if (u, v) ∈ H \ F and (e, b) ∈ F ∩ F (u,v), then there is an element
(e, w) ∈ F such that
(e, b) = (u, v)−1(e, w)(u, v)
and therefore
(e, b) = (−v−1u, v−1)(e, w)(u, v) = (v−1e − v−1u, v−1w)(u, v) =
(−v−1u, v−1w)(u, v) = (v−1wu − v−1u, v−1wv).
Since u = vi for some i ∈ I, we conclude that e = v−1wu − v−1u =
v−1wvi − v−1vi = (v−1wv − 1)i and so, in view of (1), v−1wv − 1 = 0.
But then b = 1. Hence F ∩ F (u,v) = 〈(e, 1)〉.
Corollary 3. If P is a skew-field and T is a nontrivial subgroup of the
multiplicative group P ∗, then D(P+, T ) is a Frobenius group, where P+
is the additive group of P .
Corollary 4. If G is a nontrivial abelian unitary left module over a
right nearfield N , then D(G, T ) is a Frobenius group for every nontrivial
subgroup T of the multiplicative group N∗.
As in [2, Definition 1.6.34], a nearring N is called subcommutative
if aN = Na for each a ∈ N . Recall [2, Definition 1.9.6] that a left N -
module G is called strongly monogenic if G = Ng for some g ∈ G and
for all h ∈ G it is either Nh = G or Nh = {e}. Moreover, G is faithful if
nG 6= {e} for any nonzero n ∈ N .
Proposition 5. Let G be a nontrivial faithful abelian strongly monogenic
unitary left N -module, N a subcommutative right nearring N with the
identity element 1. If T is a nontrivial subgroup of the unit group U(N),
then D(G, T ) = E ⋊ F is a Frobenius group with a kernel E ∼= G+ and
a complement F ∼= T .
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.20 Groups associated with modules over nearrings
Proof. Let us t ∈ T . If (e, t) ∈ F ∩ F (g,1) for some nonzero g ∈ G, then
(e, t) = (g, 1)−1(e, v)(g, 1)
for some element v ∈ T and from this
(e, t) = (−g, 1)(e, v)(g, 1) = (−g, v)(g, 1) = ((v − 1)g, v).
This gives that (v − 1)g = e and v = t. Since G = Ng and
(v − 1)G = (v − 1)Ng = N(v − 1)g = Ne = {e},
we obtain by the faithfulness of G that t = 1. Hence F ∩F (g,1) = 〈(e, 1)〉.
Now if h is a nonzero element of G and t is a nontrivial element of T ,
then
(t − 1)G = (t − 1)Nh = N(t − 1)h = G
and therefore
E \ {(e, 1)} = H \
⋃
(u,v)∈H
F (u,v).
A zero-symmetric right nearring N is local if NL = {k ∈ N | Nk 6=
N} is an N -subgroup [4].
Proposition 6. Let G be a nontrivial abelian monogenic unitary left
N -module, where N is a local right nearring with the identity element
1 and the zero 0 6= 1. If D(G, U(N)) is a Frobenius group, then N is a
nearfield.
Proof. By the monogenity G = Ng for some nonzero element g ∈ G.
Let j be a nontrivial element of NL. Since D(G, U(N)) is Frobenius,
we deduce that G = (1 − (1 − j))G = jG and so g = jng for some
n ∈ N . But then (1− jn)g = e. In view of Corollary 2.6 and Lemma 2.4
from [4] there exists some t ∈ N such that t(1 − jn) = 1 and therefore
g = t(1 − jn)g = te = e, a contradiction. This means that NL = {0}, as
desired.
Example 7. Let (G, +) be a group with the zero e. The set M(G) = {f :
G → G |f is a mapping} is a right nearring with the identity element iG
under two operations “+” and “◦” defined by the rules
(f1 + f2)(x) = f1(x) + f2(x) and (f1 ◦ f2)(x) = f1(f2(x))
for all elements x ∈ G and f1, f2 ∈ M(G), where f(x) means an image of
x with respect to f ∈ M(G). Hence G is an unitary left M(G)-module.
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.O. D. Artemovych, I. V. Kravets 21
1) Let G be a torsion-free divisible abelian group. If s : G → G is a
mapping defined by the rule s(g) = 2g for all g ∈ G, then sn(g) = 2ng
for all n ∈ Z and (iG − sn)(h) = (1 − 2n)h 6= e for each nonzero element
h ∈ G. Moreover, (iG−sn)(G) = G for any nonzero n ∈ Z. By Theorem 2
D(G, 〈s〉) is a torsion-free Frobenius group.
2) If f : G → G is a regular automorphism of G and G = {g −
fn(g) | g ∈ G} for any nonzero n ∈ Z, then D(G, 〈f〉) is a Frobenius
group.
3) Let G be a torsion-free abelian group, 2G = G anf t : G → G is
a mapping defined by the rule t(g) = −g for each g ∈ G. Then t2 = iG,
(1− t)(h) = h+h 6= e for any nontrivial h ∈ G and (1− t)(G) = 2G = G.
This means that D(G, 〈t〉) is a Frobenius group.
4) Let N be a distributive nearring with the identity element 1 and
P be a subfield of N with the identity element 1. Suppose that G = N+
is the additive group of N and a is a fixed element from P \ {0, 1}. Then
a mapping φ : G → G given by φ(u) = ua (u ∈ N) is an automorphism
of G and φn(u) = uan for any n ∈ Z.
If an 6= 1 for any nonzero n ∈ Z, then (iG −φn)(h) = h(1− an) = 0 if
and only if h = 0. Since 1− an ∈ P ∗, we deduce that (iG − φn)(G) = G.
Hence D(N+, 〈φ〉) is a Frobenius group.
Now suppose that an = 1 and n is the smallest positive integer with
this property. Then (iG−φs)(h) = h(1−as) 6= 0 for all nonzero h ∈ G and
integers s such that 1 ≤ s ≤ n−1. Moreover, (iG−φs)(G) = G(1−as) =
G. From this it follows that D(N+, 〈φ〉) is a Frobenius group.
References
[1] G. Pilz, Near-ring (2nd ed.), North-Holland, Amsterdam, 1983.
[2] C. Cotti Ferrero and G. Ferrero, Nearrings (Some Developments Linked to Semi-
groups and Groups), Kluwer, Dordrecht Boston London, 2002.
[3] O.D. Artemovych, On Frobenius groups associated with modules, Demonstratio
Math. 31 (1998), 875-878.
[4] C.J. Maxson, On local near-rings, Math. Zeitschrift, 106 (1968), 197-205.
Contact information
O. D. Artemovych Institute of Mathematics, Cracow Univer-
sity of Technology, ul. Warszawska 24, Cra-
cow 31155, POLAND
E-Mail: artemo@usk.pk.edu.pl
I. V. Kravets Department of Algebra and Logic, Lviv Na-
tional University of Ivan Franko, University
St 1, Lviv 79000, UKRAINE
E-Mail: yana_stud_lviv@rambler.ru
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