Download Nearrings whose set of N-subgroups is linearly ordered

Nearrings whose set of N -subgroups is linearly ordered
Peter Mayr and Fiorenza Morini
Abstract. We consider zerosymmetric right nearrings whose lattice of N -subgroups is linearly ordered by inclusion and in which for every n ∈ N , there is x ∈ N such that x ∗ n = n.
All such nearrings with finitely many N -subgroups are constructed.
1. Introduction
A right nearring (N, +, ∗) is a (2, 2)-algebra, where (N, +) is a (not necessarily abelian)
group, (N, ∗) is a semigroup, and the distributive law (a + b) ∗ c = a ∗ c + b ∗ c holds for all
a, b, c ∈ N , [1].
A subgroup H of the additive group (N, +) is an N -subgroup if N ∗ h ⊆ H for all h ∈ H.
A group (endo)automorphism α of (N, +) is an N -(endo)automorphism if α(n ∗a) = n ∗ α(a)
for all n, a ∈ N .
All nearrings in this note are zerosymmetric, that is, with 0 denoting the zero of (N, +)
we take a ∗ 0 = 0 for each a ∈ N . Moreover, |N | > 1 is assumed.
The following definition is due to A. Benini and S. Pellegrini [2].
Definition 1. A nearring (N, +, ∗) is called weakly divisible if and only if
∀a, b ∈ N ∃x ∈ N : x ∗ a = b or x ∗ b = a.
Equivalently, a nearring (N, +, ∗) is weakly divisible (wd ) if and only if the set of N subgroups of (N, +) is linearly ordered, and each element of N has a private left identity,
that is, n is in N ∗ n for each n ∈ N , ([2], Proposition 5). In particular, the set of (left)
ideals of a wd nearring is linearly ordered.
If (N, +, ∗) is an integral planar nearring [3], then N ∗ a = N for all a ∈ N, a 6= 0, and
thus (N, +, ∗) is wd.
A zerosymmetric right nearring with identity is wd if and only if its multiplicative semigroup is a left cone [4]. Every zerosymmetric wd nearring with identity is local [5] by
Lemma 1 of Section 3 below.
The N -subgroups of a ring R are exactly the left ideals of R. Hence, a ring R is wd if
and only if the set of left ideals of R is linearly ordered, and for each a ∈ R the left ideal
generated by a is equal to Ra. The wd rings with identity are exactly the left chain rings.
This note is on the characterization of zerosymmetric wd nearrings (not necessarily with
identity) with a finite chain of N -subgroups. The following construction, which resembles
2000 Mathematics Subject Classification. 16Y30.
Key words and phrases. nearrings, weakly divisible nearrings, planar nearrings, local nearrings.
The first author has been supported by Grant P12911-INF of the Austrian National Science Foundation
(Fonds zur Förderung der wissenschaftlichen Forschung).
The second author has been supported by the Italian M.U.R.S.T.
that of planar nearrings by G. Ferrero [6], and that of strongly monogenic nearrings proposed
by G. Gallina [7], is useful for this end.
Proposition 1. Let (N, +) be a group with an endomorphism ψ and with a group of automorphisms Φ, and let E ⊆ N such that the following conditions (C1), (C2), (C3) hold:
(C1) (a) Ker ψ = Im ψ r−1 for
r, r > 0, with ψ 0 := id, or
T some integer
(b) Ker ψ = {0} and i∈N Im ψ i = {0}.
(C2) Φψ ⊆ ψΦ, and Φ acts fixed-point-free on N \ Im ψ.
(That is, for each ϕ ∈ Φ there exists ϕ0 ∈ Φ such that ϕψ = ψϕ0 , and ϕ(n) 6= n for
all n ∈ N \ Im ψ, and for all ϕ ∈ Φ, ϕ 6= id.)
(C3) E is a set of orbit representatives for Φ on N \ Im ψ. Furthermore, if r is such that
Ker ψ = Im ψ r−1 , then for all i ∈ {1, . . . , r − 1}, e1 , e2 ∈ E (not necessarily distinct),
ϕ ∈ Φ we have: if ψ i ϕ(e1 ) = ψ i (e2 ), then ψ i ϕ = ψ i .
Then for all n ∈ N, n 6= 0, there are i ≥ 0, ϕ ∈ Φ, and e ∈ E such that n = ψ i ϕ(e);
furthermore, for fixed n, the endomorphism ψ i ϕ is independent of the choice of e and ϕ in
the representation of n.
For all x ∈ N, e ∈ E, ϕ ∈ Φ, and i ≥ 0 define x ∗ 0 := 0, and
x ∗ ψ i ϕ(e) := ψ i ϕ(x).
Then (N, +, ∗) is a zerosymmetric wd nearring.
We note that (C2) implies Φ(N \ Im ψ) ⊆ N \ Im ψ. Hence, if (C2) is satisfied, Φ acts on
N \ Im ψ, and thus we may speak of orbit representatives in (C3). In the sequel N, ψ, Φ, E
fulfilling conditions (C1), (C2), (C3) are called feasible. A nearring defined as in Proposition 1
for feasible N, ψ, Φ, E is denoted by W (N, ψ, Φ, E).
In this note we prove the following result.
Theorem 1. Let (N, +, ∗) be a zerosymmetric nearring with ACC and DCC on N -subgroups.
Then the following are equivalent:
(a) The nearring (N, +, ∗) is weakly divisible.
(b) The set of N -subgroups of (N, +, ∗) is linearly ordered, and n is in N ∗n for each n ∈ N .
(c) L = {k ∈ N | N ∗ k 6= N } is a proper N -subgroup, and L is an N -homomorphic image
of (N, +).
(d) There exist feasible ψ, Φ, E such that (N, +, ∗) = W (N, ψ, Φ, E).
The equivalence of (a) and (b) in Theorem 1 holds in general as stated above. The other
equivalences require the chain conditions.
The description of wd nearrings in terms of (N, +), ψ, Φ, and E immediately gives a construction method for these structures. SONATA [8] provides functions for the construction
of finite wd nearrings and geometries derived from them based on Proposition 1.
2. The construction of wd nearrings
The proof of Proposition 1 also gives the implication (d) ⇒ (a) in Theorem 1.
Proof of Proposition 1. Let n ∈ N, n 6= 0. By (C1) there exists a uniquely determined
integer i such that n ∈ Im ψ i \ Im ψ i+1 . Now n = ψ i (m) for some m ∈ N \ Im ψ. There is
e ∈ E and ϕ ∈ Φ such that m = ϕ(e). Thus we have n = ψ i ϕ(e).
If Ker ψ = {0} as in (C1) (b), then m, and thus ϕ and e in the presentation n = ψ i ϕ(e),
are uniquely determined. In particular, the mapping ψ i ϕ is unique.
If Ker ψ = Im ψ r−1 as in (C1) (a), then ψ r = 0, and ψ is nilpotent. Consequently,
Im ψ r−i ⊆ Ker ψ i for 0 ≤ i ≤ r. Let j > 1 be the least integer such that there exists
k ∈ Ker ψ j \ Im ψ r−j . Then ψ(k) is in Ker ψ j−1 , which is equal to Im ψ r−(j−1) , and there
exists n ∈ N such that ψ(k) = ψ r−(j−1) (n). The difference k − ψ r−j (n) is in Ker ψ, hence in
Im ψ r−j . Then also k is in Im ψ r−j in contradiction to the assumption. Thus Im ψ r−i = Ker ψ i
for all i with 0 ≤ i ≤ r.
We suppose that ψ i ϕ1 (e1 ) = ψ i ϕ2 (e2 ) for e1 , e2 ∈ E and ϕ1 , ϕ2 ∈ Φ. Then we have
ϕ1 (e1 ) + Ker ψ i = ϕ2 (e2 ) + Ker ψ i . As shown above Ker ψ i = Im ψ r−i , and Im ψ r−i is
i
i
invariant under automorphisms in Φ by (C2). Thus ϕ−1
2 ϕ1 (e1 ) + Ker ψ = e2 + Ker ψ , which
−1
implies ϕ2 ϕ1 (x) + Ker ψ i = x + Ker ψ i for all x ∈ N by (C3), and finally ψ i ϕ1 = ψ i ϕ2 .
This completes the proof that each n ∈ N, n 6= 0, has a presentation n = ψ i ϕ(e) with
i ≥ 0, e ∈ E, ϕ ∈ Φ, and that the endomorphism ψ i ϕ is uniquely determined by n.
Next, we prove that (N, +, ∗) is a nearring. To check associativity of ∗ we have to show
(x ∗ ψ i ϕ1 (e1 )) ∗ ψ j ϕ2 (e2 ) = x ∗ (ψ i ϕ1 (e1 ) ∗ ψ j ϕ2 (e2 ))
for all x ∈ N, e1 , e2 ∈ E, ϕ1 , ϕ2 ∈ Φ, and integers i, j. By the definition of ∗ this equation is
equivalent to
ψ j ϕ2 ψ i ϕ1 (x) = x ∗ ψ j ϕ2 ψ i ϕ1 (e1 ),
which holds, since ϕ2 ψ i can be substituted with ψ i ϕ02 for some ϕ02 ∈ Φ by (C2).
Distributivity holds by definition of the multiplication via endomorphisms. Thus (N, +, ∗)
is a nearring and is zerosymmetric by definition.
j−i
Now, let a = ψ i (ϕ1 (e1 )) and b = ψ j (ϕ2 (e2 )) with 0 ≤ i ≤ j. Then x = ϕ−1
ϕ2 (e2 )
1 ψ
solves the equation x ∗ a = b. Thus (N, +, ∗) is weakly divisible.
The ring of formal power series over a field is weakly divisible. ToPillustrate the construction of Proposition 1 we define a wd nearring on the set F [[x]] = { i≥0 ai xi | ai ∈ F } with
F a nearfield.
Example 1. P
Let (F, +, ·) P
be a nearfield, and let the addition + in F [[x]] be defined as
usual. Then ψ : i≥0 ai xi 7→ i≥0 ai xi+1 is an endomorphism of the group (F [[x]], +). Since
T
Ker ψ = {0}, and i≥1 Im ψ i = {0}, we have that ψ satisfies condition (C1) (b).
Let Φ be the group of automorphismsP
on (F [[x]], +)Pwhich are induced by multiplication
with scalars from the right, Φ = {ϕ : i≥0 ai xi 7→ i≥0 (ai · f )xi | f ∈ F }. Then each
ϕP
∈ Φ commutes with ψ, and Φ acts fixed-point-free on F [[x]], which gives (C2). Let E =
{ i≥0 ai xi | ai ∈ F, a1 = 1}. By Proposition 1 we obtain a wd nearring W (F [[x]], ψ, Φ, E) =
(F [[x]], +, ∗) where
X
X
X
ai x i ∗
bj xj :=
(ai · bk )xi+k
i≥0
and
P
i≥0
ai xi ∗ 0 := 0 for all
i≥0
j≥k
P
i≥0
ai x i ,
P
j≥k bj x
j
with bk 6= 0.
Further examples for wd nearrings, including a wd nearring which is not of the form
W (N, ψ, Φ, E), are given in Section 4.
For ψ = 0, the constant zero function, (C2) is equivalent to Φ being a group of fixed-pointfree automorphisms of (N, +). If N is finite, then ∗ in Proposition 1 is defined as according
to G. Ferrero [6], and W (N, 0, Φ, E) is an integral planar nearring.
If Φ acts fixed-point-free on the factor group N/ Im ψ, condition (C3) of Proposition 1 can
be exchanged by a more convenient one. By the next proposition it then suffices to choose E
such that {e + Im ψ | e ∈ E} is a set of orbit representatives for the action of Φ on N/ Im ψ.
Proposition 2. Let (N, +) be a group. Let ψ be an endomorphism of (N, +) with an
integer r such that Ker ψ = Im ψ r−1 . Let Φ be a group of automorphisms of (N, +) such
that Φψ ⊆ ψΦ, and such that ϕ(n) + Im ψ 6= n + Im ψ for all n ∈ N, n 6∈ Im ψ, and for all
ϕ ∈ Φ, ϕ 6= id. Let E be a set of orbit representatives for Φ acting on N \ Im ψ.
Then (C3) is satisfied if and only if {e + Im ψ | e ∈ E} is a set of orbit representatives
for Φ acting on {x + Im ψ | x ∈ N, x 6∈ Im ψ}.
Proof. For x ∈ N let x̂ = x + Im ψ. Since Im ψ is invariant under Φ by Φψ ⊆ ψΦ, there is a
group action of Φ on M = {x̂ | S
x ∈ N, x 6∈ Im ψ} with ϕ(x̂) = ϕ(x) + Im ψ for ϕ ∈ Φ.
Let E satisfy (C3). We have e∈E Φ(ê) = M . It remains to be shown that Φ(ê1 ) = Φ(ê2 )
implies ê1 = ê2 for e1 , e2 ∈ E. We assume that ϕ(ê1 ) = ê2 for ϕ ∈ Φ. Then ψ r−1 ϕ(e1 ) =
ψ r−1 (e2 ). By (C3) we obtain ψ r−1 ϕ = ψ r−1 . Now, ψ r−1 (e1 ) = ψ r−1 (e2 ) yields ê1 = ê2 , and
{ê | e ∈ E} is a set of orbit representatives on M .
Next, let {ê | e ∈ E} be a set of orbit representatives for Φ on M . We assume that
i
ψ ϕ(e1 ) = ψ i (e2 ) for e1 , e2 ∈ E and 0 < i < r. Then we have ψ r−1 ϕ(e1 ) = ψ r−1 (e2 ), or
equivalently, ϕ(ê1 ) = ê2 . This yields ê1 = ê2 . By assumption Φ has no fixed-points in M .
Now, we obtain ϕ = id, and, in particular, ψ i ϕ = ψ i . Thus (C3) is satisfied.
For N, ψ, Φ satisfying the assumptions of Proposition 2 we have Im ψ = Ker ψ r−1 . Hence
Im ψ is normal in N . Let π : N → N/ Im ψ, x 7→ x̂, and let F be a a set of orbit representatives for Φ on N/ Im ψ. Then E = π −1 (F \ {0̂}) is a set of orbit representatives for Φ on
N \ Im ψ. By Proposition 2 every set E satisfying (C3) can be obtained as pre-image under
π of some set of orbit representatives F on N/ Im ψ.
3. All wd nearrings with ACCN and DCCN
The following lemma assists in proving implication (a) ⇒ (c) of Theorem 1.
Lemma 1. Let N be a wd nearring. Then L = {k ∈ N | N ∗ k 6= N } is an N -subgroup of
N . If L 6= N , then L is the unique maximal N -subgroup of N .
Proof. Let a, b ∈ L. We assume that there exists x ∈ N such that x ∗ a = b, and we choose
y ∈ N such that y ∗ a = a. Then we have
N ∗ (a − b) = N ∗ (y ∗ a − x ∗ a) = N ∗ (y − x) ∗ a ⊆ N ∗ a 6= N,
and a − b ∈ L. If b 6∈ N ∗ a, then there exists x ∈ N such that x ∗ b = a. With z ∗ b = b the
same argumentation as above works again to show that a − b ∈ L. Thus L is a subgroup of
(N, +) and, moreover, an N -subgroup.
If N is a wd nearring with identity, then L 6= N , and N is local [5]. We note that L is not
necessarily an ideal. In Example 3 of Section 4 we will construct a simple wd nearring with
unique maximal N -subgroup.
Proof of Theorem 1, (a) ⇒ (c). Let N be a wd nearring with ACC on N -subgroups. Then
each N -subgroup H is monogenic, that is, there exists h ∈ N such that H = N ∗ h, ([2],
Theorem 5). In particular, there exists l ∈ N such that {k ∈ N | N ∗ k 6= N } = N ∗ l. Since l
is an element of N ∗ l, we have N ∗ l 6= N . The mapping ψ : x 7→ x ∗ l is an N -endomorphism
on (N, +) with Im ψ = N ∗ l. This completes the proof of (a) to (c) of Theorem 1.
The implication (c) ⇒ (d) of Theorem 1 is obtained as a corollary of the following.
Proposition 3. Let (N, +, ∗) be a zerosymmetric nearring with ACC on T
N -subgroups. Let
ψ be an N -endomorphism such that {k ∈ N | N ∗ k 6= N } = Im ψ and i≥1 Im ψ i = {0}.
Then the following hold:
(a) ψ satisfies (C1).
(b) Φ = {ϕc : x 7→ x∗c | c ∈ N \Im ψ} forms a group of automorphisms of (N, +) satisfying
(C2).
(c) The set E of right identities of (N, +, ∗) satisfies (C3).
(d) (N, +, ∗) = W (N, ψ, Φ, E).
Proof. (a) By ACC there is an integer r ≥ 1 such that
Ker ψ ⊂ Ker ψ 2 ⊂ . . . ⊂ Ker ψ r−1 ⊂ Ker ψ r = Ker ψ r+1 .
If Ker ψ ⊆ Im ψ r , then Ker ψ = ψ r (Ker ψ r+1 ) = ψ r (Ker ψ r ) = {0}, and (C1) (b) is fulfilled.
If there exists k ∈ Ker ψ \ Im ψ r , then there is c ∈ N \ Im ψ and j < r such that k = ψ j (c).
{0} = N ∗ 0 = N ∗ ψ j+1 (c) = ψ j+1 (N ∗ c) = ψ j+1 (N )
shows that ψ j+1 = 0 and Ker ψ j+1 = N = Ker ψ r . We note that r is the smallest integer such
that ψ r = 0. Necessarily, Im ψ r−1 ⊆ Ker ψ. If there exists an element k ∈ Ker ψ \ Im ψ r−1 ,
then the above argument shows that already ψ r−1 = 0. Thus Im ψ r−1 = Ker ψ, and ψ fulfills
(C1) (a).
(b) Let L = Im ψ, and let c ∈ N \ L. Then N ∗ c = N and ϕc : N → N, x 7→ x ∗ c is onto.
By ACC there is an integer k ≥ 1 such that
Ker ϕc ⊂ Ker ϕc 2 ⊂ . . . ⊂ Ker ϕc k−1 ⊂ Ker ϕc k = Ker ϕc k+1 .
Since ϕc k maps Ker ϕc k+1 onto Ker ϕc , we have Ker ϕc = (Ker ϕc k+1 ) ∗ ck = (Ker ϕc k ) ∗ ck =
{0}, and ϕc is one-to-one, thus an automorphism.
The associativity of ∗ makes Φ = {ϕc : x 7→ x ∗ c | c ∈ N \ Im ψ} a semigroup with respect
to composition. Furthermore, for c ∈ N \ L there exists a uniquely determined element
ec ∈ N such that ϕc (ec ) = c, that is, ec ∗ c = c. This implies x ∗ ec ∗ c = x ∗ c for all x ∈ N .
By canceling c we obtain x ∗ ec = x. Thus ec ∈ N \ L, and ϕec = id. The inverse of ϕc is
induced by the uniquely determined element c−1 ∈ N \ L such that c−1 ∗ c = ec . Thus Φ is
a group of N -automorphisms on (N, +).
Next we prove that Φ is fixed-point-free on N \ L. Let n, c ∈ N \ L and let ϕc (n) = n.
Then ϕc ϕn (x) = x ∗ n ∗ c = x ∗ n = ϕn (x) for all x ∈ N and ϕc = id.
To show that Φψ ⊆ ψΦ, we fix c ∈ N \L and consider (x∗ψ(ec ))∗c = ψ(x∗ec )∗c = ψ(x)∗c =
ϕc ψ(x) for x ∈ N . The same product can also be evaluated as x ∗ (ψ(ec ) ∗ c) where ψ(ec ) ∗ c is
in L. Hence there is d ∈ N such that ψ(ec )∗c = ψ(d), and x∗(ψ(ec )∗c) = x∗ψ(d) = ψ(x∗d).
By comparing (N ∗ ψ(ec )) ∗ c = ϕc (Im ψ) = Im ψ and N ∗ (ψ(ec ) ∗ c) = ψ(N ∗ d), it turns out
that ψ(N ∗ d) = Im ψ and N ∗ d = N . Thus multiplication by d induces an automorphism,
ϕd ∈ Φ, and ϕc ψ(x) = ψϕd (x) for all x ∈ N .
(c) The set E of right identities of (N, +, ∗) is formed by the elements ec = ϕc −1 (c) for
c ∈ N \ L by (b). Since c = ϕc (ec ) is an element of the orbit Φ(ec ), and since ec is the
unique right identity in this orbit, the right identities are orbit representatives for Φ acting
on N \ L.
Suppose that ψ i ϕc (e1 ) = ψ i (e2 ) for some e1 , e2 ∈ E, ϕc ∈ Φ and some integer i. Then
x ∗ ψ i ϕc (e1 ) = ψ i (x ∗ e1 ∗ c) = ψ i ϕc (x) and x ∗ ψ i (e2 ) = ψ i (x) are equal for all x ∈ N .
Therefore ψ i ϕc = ψ i , and (C3) is satisfied.
(d) Since ψ, Φ, and E as defined in (a) - (c) are feasible and x ∗ ψ i ϕ(e) = ψ i ϕ(x) for
i ≥ 0, ϕ ∈ Φ, e ∈ E, the zerosymmetric nearring (N, +, ∗) is equal to W (N, ψ, Φ, E).
The proof of Theorem 1 is now easily completed.
Proof of Theorem 1, (c) ⇒ (d). Let (N, +, ∗) be a zerosymmetric nearring with ACC and
DCC on N -subgroups, and let ψ be an N -endomorphism such that Im ψ = {k ∈ N | N ∗ k 6=
N} =
6 N .TIn order to prove (N, +, ∗) = W (N, ψ, Φ, E) for some feasible Φ and E, it suffices
to show i≥1 Im ψ i = {0}. Then Proposition 3 may be applied.
By ACC there is r ≥ 1 such that Ker ψ r = Ker ψ r+1 . By DCC there is s ≥ 1 such that
Im ψ ⊃ Im ψ 2 ⊃ . . . ⊃ Im ψ s−1 ⊃ Im ψ s = Im ψ s+1
We suppose that Ker ψ ⊆ Im ψ s . Since Im ψ s ⊆ Im ψ r , we then have Ker ψ ⊆ Im ψ r . By
Ker ψ = ψ r (Ker ψ r+1 ) = ψ r (Ker ψ r ) = {0} the endomorphism ψ is one-to-one. But with
Im ψ s = Im ψ s+1 this implies that N = Im ψ in contradiction to the assumptions.
Thus there exists k ∈ Ker ψ \ Im ψ s , that is, k = ψ j (a) with a ∈ N \ Im ψ and j < s. Now,
{0} = N ∗ 0 = N ∗ ψ j+1 (a) = ψ j+1 (N ∗ a) = ψ j+1 (N )
T
shows that ψ is nilpotent. In particular, i≥1 Im ψ i = {0}.
Not all wd nearrings with ACC on N -subgroups are of the form W (N, ψ, Φ, E). We next
give such an example.
Example 2. Let Z2 [[x]] denote the set of formal power series over the field with 2 elements.
We define ∗ on N = Z2 [[x]] × Z2 such that
X
(
ai xi , b) ∗ (0, d) := (0, a0 d), and
i≥0
X
X
X
(
ai xi , b) ∗ (
cj xj , d) := (
ai xi+k , b)
i≥0
j≥k
i≥0
for all i≥0 ai xi , j≥k cj xj ∈ Z2 [[x]] with ck = 1 and for all b, d ∈ Z2 .
This operation is associative, and the right distributive law for + and ∗ holds. Thus
(N, +, ∗) is a nearring. Its non-trivial N -subgroups are given by N ∗ li with l := (x, 0) and
i ≥ 1 and by H = {(0, d) | d ∈ Z2 }. Hence, (N, +, ∗) is wd and satisfies the ACC on
N -subgroups but not the DCC.
By Theorem 2 (a) of Section 4 all non-trivial N -subgroups of a nearring W (N, ψ, Φ, E) are
of the form Im ψ i . Hence, W (N, ψ, Φ, E) has a finite number of N -subgroups if ψ satisfies
condition (C1)(a), and the intersection of all non-trivial N -subgroups yields {0} if ψ satisfies
(C1)(b) of Proposition 1.
Now, (N, +, ∗) has infinitely many non-trivial N -subgroups, N ∗ li for i ∈ N and H, and
their intersection yields H 6= {(0, 0)}. Thus (N, +, ∗) is not of the form W (N, ψ, Φ, E).
P
P
4. The Structure of W (N, ψ, Φ, E)
If a wd nearring (N, +, ∗) has a representation of the form W (N, ψ, Φ, E), then the structure of (N, +, ∗) can be described nicely in terms of ψ, Φ, E.
Theorem 2. Let (N, +, ∗) = W (N, ψ, Φ, E). Then we have:
(a) The N -subgroups of N are given as Im ψ i with i ≥ 0 and {0}.
(b) Im ψ = {k ∈ N | N ∗ k 6= N } =
6 N.
(c) E is the set of right identities of (N, +, ∗).
(d) (N, +, ∗) has the identity e if and only if E = {e}.
Proof. (a) Each Im ψ i for i ≥ 0 is a subgroup of (N, +). Moreover, x ∗ ψ i (n) = ψ i (x ∗ n) ∈
Im ψ i for all x, n ∈ N by the definition of the multiplication. Thus Im ψ i is also an N subgroup.
Let H 6= {0} be an N -subgroup, and let i ≥ 0 be maximal such that H ⊆ Im ψ i . Then
there exists h ∈ H with h ∈ Im ψ i \ Im ψ i+1 , that is, h = ψ i ϕ(e) for some e ∈ E, ϕ ∈ Φ. By
H ⊇ N ∗ h = ψ i ϕ(N ) = Im ψ i we obtain Im ψ i = H.
(b) Since Im ψ 6= N is the unique maximal N -subgroup of the wd nearring N by (a), we
have Im ψ = {k ∈ N | N ∗ k 6= N } by Lemma 1.
(c) All orbit representatives E are right identities for (N, +, ∗) by definition of the multiplication, x ∗ id(e) = id(x) = x for all x ∈ N, e ∈ E.
Let n = ψ i ϕ(e) be a right identity for (N, +, ∗). Then we have i = 0, and e = e ∗ n =
e ∗ ϕ(e) = ϕ(e). By condition (C2) ϕ(e) = e implies ϕ = id, and finally n = e ∈ E.
(d) Let e be the identity of (N, +, ∗). Then e ∈ E by (c). For e0 ∈ E we obtain
e = e ∗ e0 = e0 . Thus E = {e}.
If E = {e}, then for every element n ∈ N, n 6= 0, there is i ≥ 0, and there is ϕ ∈ Φ such
that n = ψ i ϕ(e). Thus e ∗ n = e ∗ ψ i ϕ(e) = ψ i ϕ(e) = n, and n ∗ e = n by (c).
For a nilpotent endomorphism ψ stronger results on W (N, ψ, Φ, E) can be obtained. As
shown in the proof of Proposition 1 the group (N, +) has then a normal series
N = Im ψ 0 Im ψ 1 Im ψ 2 · · · Im ψ r−1 Im ψ r = {0},
where for all 0 ≤ i ≤ j ≤ r the factor Im ψ i / Im ψ j is isomorphic to Im ψ r−(j−i) .
Corollary 1. Let (N, +, ∗) = W (N, ψ, Φ, E), and let ψ be nilpotent with r minimal such
that ψ r = 0. Then we have:
(a) The set of ideals of N is equal to {Im ψ i | 0 ≤ i ≤ r}.
(b) Im ψ is a prime ideal, and the elements of Im ψ are nilpotent.
(c) Im ψ is the prime radical of N , and Im ψ = J2 (N ).
(d) N/ Im ψ is an integral nearring.
Proof. (a) By the proof of Proposition 1 Im ψ i is the kernel of ψ r−i and thus an ideal for
0 ≤ i ≤ r. By Theorem 2 (a) there are no further N -subgroups, hence no further ideals.
(b) Im ψ is the unique maximal ideal of N , and it is prime since N ∗ N = N . For each
element ψ(n) ∈ Im ψ we have (ψ(n))r ∈ Im ψ r = {0}, and hence (ψ(n))r = 0.
(c) The ideal product Im ψ i−1 ∗ Im ψ is equal to Im ψ i but neither Im ψ i−1 nor Im ψ are in
Im ψ i for 2 ≤ i ≤ r. Thus Im ψ i with i > 1 is not prime. The prime radical of N , defined as
the intersection of all prime ideals of N , is then equal to the unique prime ideal Im ψ.
The Jacobson radical J2 (N ) is the intersection of all 2-modular left ideals of N . The left
ideals are given as the N -subgroups. N is 2-primitive on N/ Im ψ i if and only if Im ψ i = Im ψ,
and since there exists a right identity e ∈ E of N , Im ψ is the unique 2-modular left ideal of
N . Hence, J2 (N ) = Im ψ.
(d) Suppose that (a + Im ψ) ∗ (b + Im ψ) = Im ψ. Since a ∗ b ∈ Im ψ implies a ∈ Im ψ or
b ∈ Im ψ, there are no zero-divisors in N/ Im ψ.
In the following example we determine all wd nearrings with additive group (Z, +).
Example 3. Let (Z, +, ∗) be a wd nearring with + denoting the usual addition on Z.
The subgroups of (Z, +) satisfy the ACC and the same holds for the N -subgroups. Let
L = {k ∈ Z | Z ∗ k 6= Z} be the maximal proper N -subgroup. By ([2], Theorem 5) there is
l0 ∈ Z such that L = Z ∗ l0 , that is, L is the image of the N -endomorphism ψ : x 7→ x ∗ l0 .
Now, all group-endomorphisms of (Z, +) are of the form x 7→ ax for some a ∈ Z. Thus
there is l ∈ Z suchTthat ψ(x) = lx for all x ∈ Z. Since lZ = L 6= Z, we have l 6∈ {−1, 1},
and, furthermore, i≥0 li Z = {0}. The assumptions of Proposition 3 hold, and there are
feasible Φ, E such that (Z, +, ∗) = W (Z, ψ, Φ, E).
Next, we determine the feasible parameters ψ, Φ, E on (Z, +). The endomorphism
ψ : x 7→
T
i
lx with l ∈ Z, l 6∈ {−1, 0, 1}, on (Z, +) fulfills (C1) (b), since Ker ψ = {0} and i≥0 l Z = {0}.
For ψ = 0 condition (C1) (a) holds, since Ker ψ = Im ψ 0 with ψ 0 = id.
The only group automorphisms of (Z, +) are id and − id. Both commute with any ψ, and
both Φ1 = {id} and Φ2 = {id, − id} are fixed-point-free on Z. Thus Φ1 and Φ2 satisfy (C2).
The multiplication ∗ of W (Z, ψ, {id}, Z \ lZ) can be expressed in terms of the usual ring
multiplication. We have x ∗ eli = xli and x ∗ 0 = 0 for all x ∈ Z, e ∈ Z \ lZ and i ≥ 0. The
set of N -subgroups is equal to the set of ideals, namely, {li Z | i ≥ 0} ∪ {{0}}.
For Φ2 = {id, − id} we may choose for example E = N \ lZ. Then x ∗ eli := xli and
x ∗ (−eli ) := −xli for e ∈ E. The nearring W (Z, ψ, {id, − id}, N \ lZ) with l > 2 is simple,
since none of the N -subgroups is a left ideal. For li ∈ li Z consider x ∗ (−1 + li ) − x ∗ (−1) =
x − (−x) = 2x which is not in li Z for arbitrary x ∈ Z.
If (N, +, ∗) is a finite wd nearring, then it is of the form W (N, ψ, Φ, E), where Φ splits.
Proposition 4. Let (N, +, ∗) = W (N, ψ, Φ, E) be finite with r minimal such that ψ r = 0.
Then H = {ϕ ∈ Φ | ψ r−1 ϕ = ψ r−1 } is a normal subgroup of Φ; there exists a complement A
for H in Φ such that Φ = H · A, and A acts fixed-point-free on N/ Im ψ.
Proof. For ϕ ∈ Φ let ϕ̂ : N/ Im ψ → N/ Im ψ, x̂ 7→ ϕ(x̂). Then h : ϕ 7→ ϕ̂ is a group
homomorphism mapping Φ to a group of automorphisms Φ̂ of N/ Im ψ. Since Im ψ =
Ker ψ r−1 , we have Ker h = {ϕ ∈ Φ | ψ r−1 ϕ = ψ r−1 } = H, and H is a normal subgroup of
Φ. Moreover, since Φ has no fixed points in N \ Im ψ by (C2), Φ̂ acts fixed-point-free on
N/ Im ψ, and |Φ̂| divides |N/ Im ψ| − 1.
Now, the set e+Im ψ is invariant under Ker h = H, and H has no fixed-points in e+Im ψ ⊆
N \ Im ψ. Thus e + Im ψ is the union of equal-sized orbits of H, and |H| divides | Im ψ|.
Let q denote the size of Ker ψ. Then | Im ψ| = q r−1 and |N/ Im ψ| − 1 = q − 1. The sizes
of Φ̂ and H are relatively prime. By the Schur-Zassenhaus Theorem ([9], 9.1.2) there exists
a subgroup A of Φ isomorphic to Φ̂ such that Φ is a semidirect product of H and A.
Theorem 2 in [10] determines the wd nearrings on (Zpr , +) with p prime. In the following
example we give the feasible choices for ψ, Φ, and E for the construction of wd nearrings on
(Zn , +) where n is not a prime power.
Example 4. Let (Zn , +, ∗) be a wd nearring, that is, of the form W (N, ψ, Φ, E) by
Theorem 1. Let the maximal N -subgroup Im ψ be equal to qZn with q dividing n. Then
there exists l coprime to q such that ψ : x 7→ lqx and n = q r where r is minimal such that
ψ r = 0 by (C1) (a).
Let Φ = {ϕ : Zn → Zn , x 7→ x ∗ c | c ∈ Zn \ qZn }. We suppose that ϕ ∈ Φ, ϕ 6= id,
has a fixed point distinct from 0. The set of fixed points of ϕ forms an N -subgroup and
therefore equals q s Zqr with 0 < s < r by Theorem 2 (a). Let ϕ(x) = f x with f ∈ Z. Then
gcd(f − 1, q r ) = q r−s . Now, for q not prime let d, 1 6= d 6= q, be a divisor of q, and consider
ϕd (x) = f d x. A number theoretical consideration yields a contradiction:
fd − 1
= f d−1 + · · · + f + 1
f −1
≡ 1| + ·{z
· · + 1} mod q
d times
≡ d mod q
Consequently, gcd(f d − 1, q r ) = dq r−s , and the set of fixed points of ϕd ∈ Φ is equal to
qs
Z r , which is not an N -subgroup since 1 6= d 6= q. Thus Φ is a group of fixed-point-free
d q
automorphisms.
Conversely, (Zqr , +) with ψ : x 7→ lqx and l coprime to q, Φ a group of fixed-point-free
automorphisms, and E as given by Proposition 2 is feasible for all q and gives a wd nearring
W (Zqr , ψ, Φ, E).
In the following theorem we determine which “inputs” give rise to isomorphic nearrings
W (N, ψ, Φ, E).
Theorem 3. The nearrings W (N1 , ψ1 , Φ1 , E1 ) and W (N2 , ψ2 , Φ2 , E2 ) are isomorphic if and
only if the following conditions are satisfied:
(a) There exists a group isomorphism α from (N1 , +1 ) to (N2 , +2 ).
(b) E2 = α(E1 ).
(c) Φ2 = αΦ1 α−1 .
(d) ψ2 ∈ αψ1 Φ1 α−1 .
Proof. Let the multiplication in W1 = W (N1 , ψ1 , Φ1 , E1 ) be denoted with ∗1 and the multiplication in W2 = W (N2 , ψ2 , Φ2 , E2 ) with ∗2 .
“⇒”: (a),(b) Let α be a nearring isomorphism from W1 to W2 . Then, in particular, α is a
group isomorphism from (N1 , +) to (N2 , +). Necessarily, α maps the right identities of W1
to the right identities of W2 , thus α(E1 ) = E2 .
(c) For all ϕ1 ∈ Φ1 and e1 ∈ E1 the equality α(x ∗1 ϕ1 (e1 )) = α(x) ∗2 αϕ1 (e1 ) holds and is
equivalent to αϕ1 (x) = α(x) ∗2 αϕ1 (e1 ). Now, αϕ1 (e1 ) 6∈ Im ψ2 . Thus αϕ1 (e1 ) = ϕ2 (e2 ) for
some ϕ2 ∈ Φ2 , e2 ∈ E2 and αϕ1 (x) = ϕ2 α(x) for all x ∈ N . Finally, αΦ1 α−1 ⊆ Φ2 , and the
converse inclusion follows from the fact that α−1 is a nearring isomorphism from N2 to N1 .
(d) We consider α(x ∗1 ψ1 (e1 )) = α(x) ∗2 αψ1 (e1 ) for e1 ∈ E1 . Now, α maps the maximal
N -subgroup of W1 to that of W2 , that is α(Im ψ1 ) = Im ψ2 . With αψ1 (e1 ) = ψ2 ϕ2 (e2 ) for
some ϕ2 ∈ Φ2 , e2 ∈ E2 we then obtain αψ1 (x) = ψ2 ϕ2 α(x) and ψ2 ∈ αψ1 Φ1 α−1 .
“⇐”: Let (a) to (d) hold. It suffices to show that
α(x ∗1 ψ1i ϕ(e)) = α(x) ∗2 αψ1i ϕ(e)
for all x ∈ N1 , ϕ ∈ Φ1 , e ∈ E1 , i ≥ 0. Let ϕ2 ∈ Φ2 such that ψ1 = α−1 ψ2 ϕ2 α. Then
α(x) ∗2 αψ1i ϕ(e) = α(x) ∗2 α(α−1 ψ2 ϕ2 α)i ϕ(e)
= α(x) ∗2 (ψ2 ϕ2 )i (αϕα−1 )α(e)
Since αϕα−1 ∈ Φ2 and α(e) ∈ E2 , this yields
α(x) ∗2 αψ1i ϕ(e) = (ψ2 ϕ2 )i (αϕα−1 )α(x)
= (αψ1 α−1 )i (αϕα−1 )α(x)
= αψ1i ϕ(x)
= α(x ∗1 ψ1i ϕ(e))
which completes the proof.
Acknowledgements. The authors wish to thank Erhard Aichinger for his assistance in
the preparation of the final form of this note.
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Peter Mayr,
Institut für Mathematik,
Johannes Kepler Universität,
A-4040 Linz, Austria.
Email: [email protected]
Fiorenza Morini,
Dipartimento di Matematica
- Facoltá di Ingegneria Universitá degli Studi di Brescia,
Via Valotti 9, 25133 Brescia, Italia.
Email: [email protected]
Eingegangen am 18. März 2003