LOCAL NEAR-RINGS WITH COMMUTATIVE GROUPS OF UNITS

ALEXANDER GORODNIK

Abstract. Properties of a local near-ring with a commutative group of units are studied in the paper. For this purpose, a method of reduction to the radical ring is introduced. It is proven that the additive group of such nearring will also have some commutative properties. Several known results about groups of units of rings are extended to the groups of units of local near-rings. Specifically, the equivalence of finite generation for the additive group and the multiplicative group of a local near-ring is established, and a comprehensive classification of local near-rings with cyclic groups of units is given.

1. Introduction

One of the common questions in the theory of rings is the mutual relation between a ring and its group of units. To put it more exactly, two problems arise naturally:

(1) Determine the group of units of a given ring.

(2) Characterize rings with certain groups of units. Partial cases of this situation were considered by many authors, and fundamental results have been obtained (see [5], [7, Ch. 18], and [10]). Nonetheless, very little is known about the groups of units of near-rings in spite of the growing interest in the subject during the recent years. The objective of this research is to develop a new method for near-ring investigation and to clarify interrelations between near-rings and their groups of units by applying this method.

In this paper, the notion “near-ring” means a right near-ring (i. e. with the distributive law (x + y)z = xz + yz) with identity. Some basic definitions and results regarding near-rings can be found in [14]. We denote the additive group of S by S^{+} and the group of units of S by S^{∗}. Our exposition will be limited to the local near-rings. This concept was introduced by Maxson in [11]. He defines a local near-ring as the near-ring N where the set L of elements without left inverses forms a N-subgroup. Apparently, if N is a ring, it will be a local ring in the usual sense. Maxson also proves there that the group of units N^{∗} coincides with the complement set N\L. Elements in L have no right inverses, and L is a (N, N)-subgroup but not always an ideal. The set L is quasi-regular. We will use these facts without any reference.

The first of the stated problems seems to be easier. The structure of multiplicative groups of several well-known near-rings is given in [2, Ch. 5]. In particular, the group of units of the near-ring (F [x], +, ∘) where F is a field is described. Note that this near-ring is local when |F | > 2. However, there is no result concerning the structure of near-rings with given groups of units. The case when the group of units is abelian will be presented here.

1991 Mathematics Subject Classification. 16Y30, 16L99, 16U60, 20D40.

It should be noted that the group of units does not determine its near-ring in a general case. This group can be an identity group for many nontrivial near-rings. Hence, we need to impose additional conditions on the near-ring. In particular, the group of units should be “large” enough. For this reason, we confine our study to the local near-rings.

The structure of the additive group of a near-ring satisfying mentioned conditions is described at the first step (Theorem 1). This group has certain commutative properties. Then, the relationship between the additive group and the group of units is derived for the case if one of them is finitely generated (Theorems 2). Using it, we will find that the near-ring is finite in this case (Theorem 3). As an application of the obtained results, a complete classification of the local near-rings with cyclic groups of units will be given in conclusion (Theorem 4).

The principal method of this paper is formulated in Proposition 1. Due to it, some questions concerning local near-rings can be reformulated for the triply factorized groups. One can find exhaustive information about triply factorized groups in [1, Chapter 6]. A group G will be called triply factorized if it can be written as G = AB = AM = BM where the subgroup M is normal in G and A ∩ B = A ∩ M = B ∩ M = 1. A group G has this form if and only if there exists a derivation ϕ from A onto M, i. e. a surjective map with the property ϕ(xy) = ϕ(x)^{y}ϕ(y) for every x, y from A. If A, B, and M are abelian, a radical ring (R, +, ×) exists for which the additive group R^{+} = M, the quasi-regular group R^{∘} = A and m^{a} = m + m × a for m ∈ M, a ∈ A. Applying the theory of triply factorized groups, we will define from the local near-ring a radical ring that determines the multiplication table x · (1 + y) where x and y have no right inverse. This observation fulfills the crucial part of the proof. Using known results about the quasi-regular group of a radical ring, we will be able to deduce the major properties of the whole near-ring.

The mentioned notations will be used throughout this paper. Also denote by C_{n} the cyclic group of order n and by Z(G) the centre of a group G.

2. Main results

The following theorem shows a nontrivial relation between the multiplicative group and the additive group in a local near-ring.

Theorem 1. Let N be a local near-ring with a commutative group of units. Then the subgroup L of elements without left inverses is an ideal and one of the following holds:

(1) The additive group N^{+} is abelian.

(2) The additive group L^{+} is abelian and has index 2 in N^{+}. In particular, N/L is a field.

Example. Build an example of a finite non-abelian local near-ring satisfying the second case of Theorem 1. Note that the additive group of this near-ring must be a 2-group [11], and neither the dihedral group [12] nor the quaternion group [4] can be the additive group of a local near-ring. Therefore, this example has a cardinality at least 16. The additive group of the desired near-ring is defined in terms of generators and relations:

N^{+} = ⟨1, x | 4 · 1 = 4 · x = 0, −1 + x + 1 = −x⟩ .

Let L = ⟨2⟩ ⊕ ⟨x⟩ ≤ N. Multiplication will be defined as double-sided distributive on L and in addition

x · x = 2x, l_{1} · (1 + l_{2}) = l_{1} for l_{1}, l_{2} ∈ L

It can be checked directly that N is a local near-ring and N^{∗} = (1 + L)^{∗} ≃ L^{+} ≃ C_{2} ⊕ C_{4}.

It is well known that for a radical ring the additive group is finitely generated if and only if the quasi-regular group is finitely generated [5, Theorem 1]. Similar results for local near-rings also must be true.

Theorem 2. Let N be a local near-ring with a commutative group of units, and L is the subgroup of elements without left inverses.

Then the following conditions are equivalent:

(α) N^{+} is finitely generated.

(β) L^{+} is finitely generated.

(γ) (1 + L)^{∗} is finitely generated. (δ) N^{∗} is finitely generated.

Artinian rings with nilpotent groups of units were studied in [9]. The exhaustive characterization is given in this paper. In particular, every such ring must be finite. We will prove that this is also true for local near-rings when the multiplicative group is finitely generated abelian.

Theorem 3. If N is a local near-ring with a finitely generated commutative group of units, then N is finite.

Since the subgroup L in Theorem 2 is quasi-regular and finite by the last theorem, it is nilpotent (see [14, Theorem 3.40]), This remark corresponds to the second part of [5, Theorem 1].

Gilmer in [8] classified all finite rings that have cyclic groups of units. An analogous result for local near-rings is presented here. Strictly speaking, it is not a generalization. However, note that the first step in Gilmer’s proof is a reduction to the local ring case.

Because all rings have already been listed in [8], only nontrivial near-rings (that are not rings) are given explicitly.

Theorem 4. Let N be a local near-ring with a (finite or infinite) cyclic group of units. Suppose that N is not a ring.

Then N can be one of the following:

• NRC_{4,2}: N^{+} = ⟨1⟩ ⊕ ⟨u⟩, 4 · 1 = 2 · u = 0, u · u = u · 2 = u · (2 + u) = 0, u · (1 + u) = u · (3 + u) = u + 2,

u · 3 = u.

• NRC_{2,2,2}: N^{+} = ⟨1⟩ ⊕ ⟨u_{1}⟩ ⊕ ⟨u_{2}⟩, 2 · 1 = 2 · u_{1} = 2 · u_{2} = 0, u_{i} · u_{j} = u_{i} · (u_{1} + u_{2}) = 0, u_{i} · (1 + u_{1} + u_{2}) = u_{i},

u_{1} · (1 + u_{i}) = u_{2}, u_{2} · (1 + u_{i}) = u_{1} where i = 0, 1 and j = 0, 1.

Note that in any case L has a zero multiplication.

3. Proofs of the theorems

A proof of Theorem 1 will be divided into several lemmas. We assume implicitly in their formulations that a local near-ring N has an abelian group of units, and L is the subgroup of elements of N without left inverses.

Lemma 1. If r_{1}, r_{2}, r_{1} + r_{2} ∈ N^{∗}, then r_{1} + r_{2} = r_{2} + r_{1}.

Proof. (r_{2} + r_{1}) · (−1) = r_{2} · (−1) + r_{1} · (−1) = (−1) · r_{2} + (−1) · r_{1} = −r_{2} − r_{1} = (−1) · (r_{1} + r_{2}) = (r_{1} + r_{2}) · (−1). Also (−1)^{2} = −(−1) = 1. Multiplying right side by (−1), we obtain the required equality. □

Lemma 2. If r ∈ N^{∗} and l ∈ L, then −r + l + r = −l · (−1).

Proof. (−1)·(l+r) = −(l+r) = −r−l = (−1)·r−l = r·(−1)−l = (r−l·(−1))·(−1) = (−1)·(r−l·(−1)). Multiplying left side by (−1), we obtain the required equality. □

For any r from N, denote r′ = −r · (−1).

Lemma 3. For any r from N, (r′)′ = r.

Proof. (r′)′ = −(−r · (−1)) · (−1) = r · (−1) · (−1) = r. □

Lemma 4. L^{+} is commutative.

Proof. Let l_{1}, l_{2} be in L. Applying Lemma 2 twice, we obtain that

(1 + l_{1}) + l_{2} = l′_{2} + (1 + l_{1}) = (1 + l_{2} − 1) + 1 + l_{1} = 1 + l_{2} + l_{1}.

Thus, L^{+} is abelian.^{1} □

Lemma 5. If L^{+} ≤ Z(N^{+}), then N^{+} is commutative.

Proof. The case when |N^{+} : L^{+}| = 2 is obvious. Suppose that |N^{+} : L^{+}| > 2. Let elements r and s be in N^{∗}. If r + s ∈ N^{∗}, r + s = s + r by Lemma 1. Now assume that r + s ∈ L. There exists an element r_{1} from N^{∗} such that r + r_{1} ∈ N^{∗}. By Lemma 1, r + r_{1} = r_{1} + r. Thus,

(1)  r + r_{1} + s = r_{1} + (r + s) = (r + s) + r_{1}

by assumption. Hence, r_{1} and s also commute. The element r_{1} + (r + s) lies in N^{∗}. Again by Lemma 1,

(2)  (r_{1} + r) + s = s + (r_{1} + r) = s + r + r_{1}

Combining (1) and (2), obtain that r + s = s + r. □

Proof of Theorem 1. L^{+} is abelian by Lemma 4.

At first, assume that |N^{+} : L^{+}| > 2. There exist elements r_{1} and r_{2} from N^{∗} such that r_{1} + r_{2} ∈ N^{∗}. By Lemmas 2 and 3,

l′ = −(r_{1} + r_{2}) + l + (r_{1} + r_{2}) = −r_{2} + (−r_{1} + l + r_{1}) + r_{2} = (l′)′ = l.

for every l from L. This means that L^{+} ≤ Z(N^{+}) and N^{+} is commutative by Lemma 5. Therefore, the additive group N^{+} is abelian or |N^{+} : L^{+}| = 2. By any way, L^{+} is normal in N^{+}.

Now prove that L is an ideal. We should show that for every r, s from N and l from L, the element r · (s + l) − r · s lies in L. If r or s lies in L, this follows from the fact that L is a (N, N)-subgroup. In the other case, from the commutative condition r · (s + l) − r · s = (s + l) · r − s · r = s · r + l · r − s · r ∈ L. □

The following proposition plays crucial rule in the proofs of Theorems 2 – 3 and in the classification of the near-rings with a cyclic group of units.

1This proof was simplified by the referee.

Proposition 1. Let N be a local near-ring whose group of units is abelian and L is the subgroup of elements without left inverses. Then, a commutative radical ring (L, +, ×) (with respect to a new multiplication “×”) exists such that for any s_{1} and s_{2} from L,

(3)  s_{1} · (1 + s_{2}) = s_{1} + s_{1} × s_{2}.

Proof. L^{+} is abelian by Theorem 1. Let a group G = L^{+} ⋊ (1 + L)^{∗} be the semidirect product where (1 + L)^{∗} acts on L^{+} by right multiplication. Define the map ϕ : (1 + L)^{∗} → L^{+} by the rule ϕ(1 + s) = s for every s from L. Then for s_{1} and s_{2} from L,

ϕ((1 + s_{1}) · (1 + s_{2})) = ϕ(1 + s_{2} + s_{1} · (1 + s_{2})) = s_{2} + s_{1}·
·(1 + s_{2}) = ϕ(1 + s_{1}) · (1 + s_{2}) + ϕ(1 + s_{2}).

Hence, ϕ is a derivation and, according to [15], there exists a subgroup H of G with the following properties:

(4) G = L^{+} ⋊ (1 + L)^{∗} = L^{+} ⋊ H = (1 + L)^{∗}H and (1 + L)^{∗} ∩ H = 1.

Consequently, the group G is triply factorized by abelian subgroups and a required radical ring can be found by [1, Proposition 6.1.4].

The quasi-regular group of the constructed radical ring is isomorphic to (1+L)^{∗}, so that this ring must be commutative. □

Proof of Theorem 2. The facts that (δ) ⇒ (γ) and (α) ⇒ (β) follow from the condition on N^{∗} and the structure of N^{+} given by Theorem 1.

(β) ⇒ (α). Consider the homomorphism U_{l} : N^{+} → L^{+}, U_{l}(r) = r · l defined for any l from L. We have that Ker U_{l} ≤ L. Hence, N^{+}/L^{+} can be embedded in a finitely generated abelian group. Therefore, N^{+}/L^{+} is finitely generated and so is N^{+}.

(γ) ⇔ (β). It follows from Proposition 1 that a radical ring R exists with the additive group R^{+} ≃ L^{+} and the quasi-regular group R^{∘} ≃ (1 + L)^{∗}. Hence, this case may be obtained directly from [5, Theorem 1].

(α) ⇒ (δ). According to Theorem 1, N/L is a field. Because (N/L)^{+} is finitely generated, it must be finite. Also (1 + L)^{∗} is finitely generated by the previous arguments. Consequently, N^{∗} is finitely generated. □

Proof of Theorem 3. Let N be a local near-ring with a commutative finitely generated group of units. L is the subgroup of elements of N without left inverses. By Theorem 2, N^{+} and L^{+} are finitely generated. It follows from Theorem 1 that N/L is a field. Because the group (N/L)^{+} is finitely generated, N/L must be finite.

Define the radical ring R as in the proof of Theorem 2. R is a nilpotent ring where the additive group is isomorphic to L^{+}, and the quasi-regular group is isomorphic to (1 + L)^{∗}. As in the proof of Proposition 1, we can consider a group G of the form (4). Since R is nilpotent, and G is defined by the rule (3), G is nilpotent too. Next, from relation (3)

Z(G) = (Ann R)(1 + Ann R),

where Ann R is the annihilator of the ring R.

At first, consider the case when N^{+} is abelian. Because N/L is finite, k · 1 lies in L for some natural k. Let y ∈ Ann R. Then

k · 1 = (k · 1)(1 + y) = k · (1 + y) = k · 1 + k · y.

Hence, k · y = 0 for every y from Ann R. Since (Ann R)^{+} is finitely generated, it must be finite. Therefore, Z(G) is finite. But then G is finite by [13, Theorem 2.24]. In particular, L is finite and so is N.

If N^{+} is not abelian, |N : L| = 2. Thus, 2 · 1 ∈ L. We have x · (1 + y) = x for every y from Ann R and any x from L. In particular, 2 · (1 + y) = 2 where y ∈ Ann R. Multiplying by x from L, obtain 2 · (x + yx) = 2x. Consequently, 2 · yx = 0 and 2 · Ann R ≤ Ann_{l} L = (0 : L). Note that Ann_{l} L is an ideal of N. We conclude the proof by induction on the torsion-free rank r_{0} of L^{+}. Recall that the torsion-free rank of an abelian group is the cardinal of a maximal independent set consisting of elements with infinite order. If r_{0}(L^{+}) = 0, then L is finite and so is N. Let r_{0}(L^{+}) > 0. Suppose that r_{0}((Ann_{l} L)^{+}) = 0. Then (Ann R)^{+}/(Ann R ∩ Ann_{l} L)^{+} is an elementary abelian 2-group and it must be finite. Then Ann R is also finite, and by [13, Theorem 2.24], G is finite. This is a contradiction. Hence, r_{0}((L/(Ann_{l} L))^{+}) < r_{0}(L^{+}). By the inductive assumption, N/(Ann_{l} L) must be finite. Then k · 1 ∈ Ann_{l} L for some natural number k. It follows that (2k) · 1 = (k · 1)(2 · 1) = 0. Therefore, N^{+} has a finite exponent, and consequently, it is finite. □

Actually, it was sufficient for the proof that the subgroup (1 + L)^{∗} was finitely generated.

This lemma complements Theorem 3 from [3] slightly.

Lemma 6. Let N be a near-ring with identity. Then the (additive) order of all elements that have right (multiplicative) inverses is equal to the order of identity.

Proof. It is proven in [3] that identity has the maximal order in N^{+}. Let m be an element with right inverse n and k · m = 0 for some natural number k. Then

k · 1 = k · (m · n) = (k · m) · n = 0,

and lemma results follow. □

The following technical result will be useful later.

Lemma 7. Let N be a local near-ring. Then for any x and y which are not units, x · y = y implies y = 0.

Proof. Suppose that x · y = y for some elements x and y without inverses. Then x − 1 is a unit and (x − 1) · y = 0. Thus, y = 0. □

The classification of artinian radical rings with a cyclic quasi-regular group from [6] will be used along the proof.

Proof of Theorem 3. The case when N^{∗} ≃ C_{∞} can be eliminated by Theorem 3. Hence, no local near-ring has C_{∞} as its group of units.

Assume now that N^{∗} is a finite cyclic group. Then N is also finite. According to [11], the additive group of N is a p-group for some prime number p. Cases p > 2 and p = 2 will be analyzed separately:

(1) p > 2. Theorem 1 states that N^{+} is abelian in this case. By Proposition 1, a radical ring (L, +, ×) exists with the quasi-regular group L^{∘} isomorphic to (1 + L)^{∗}. From [6], obtain

(5)  (L, +, ×) = {ku | k = 0, … p^{n} − 1, u × u = p^{i}u},

for some i = 1, … n. Thus, L^{+} is cyclic of order p^{n} and

(ku) · (1 + k_{1}u) = (k + kk_{1}p^{i})u.

Look at the homomorphism U_{l} : N^{+} → L^{+}, U_{l}(r) = r · l that is defined for an arbitrary non-zero l from L. Clearly, Ker U_{l} ≤ L and (N/Ker U_{l})^{+} embeds in L^{+}. But L^{+} is cyclic, and (N/L)^{+} is elementary abelian because N/L is a finite field. Therefore, |N/L| = p. For some k_{0} = 0, … p^{n}−1, p· 1 = k_{0}u. If k_{0} is coprime with p, N^{+} is cyclic and N is a ring by [3]. In the other case p·1 = p·u_{1} for some u_{1} from L. Because N^{+} is abelian, we obtain that p · (1 − u_{1}) = 0. Then p · 1 = 0 by Lemma 6, and p · x = (p · 1) · x = 0 for any x from N. Hence, |L| = p. Since p is prime, the homomorphism U_{l}|_{L} is either surjective or zero. In the first case, s · l = l for some s from L. By Lemma 7, l = 0. This is a conradiction. Consequently, U_{l}|_{L} is the zero map for any l in L, i. e. L has zero multiplication. Finally, (k · 1) · (1 + l) = (1 + l) · (k · 1) for k = 1, … p − 1 and l from L. Therefore (k · 1) · l = l · (k · 1), and N is a commutative ring. It follows that N is the ring Z_{p}[x]/ ⟨x^{2}⟩.

Thus, there is no nontrivial local near-ring when p > 2.

(2) p = 2. Applying Proposition 1, we introduce a radical ring (L, +, ×) such that l_{1} · (1 + l_{2}) = l_{1} + l_{1} × l_{2} for l_{1}, l_{2} ∈ L. According to [6], this ring has the form (5) with p = 2 or

(L, +, ×) = {0, u_{1}, u_{2}, u_{1} + u_{2} | 2u_{1} = 2u_{2} = 0, u_{1} × u_{1} = (6)

= u_{1} × u_{2} = u_{2} × u_{2} = u_{2} × u_{1} = u_{1} + u_{2}}.

Consider first the case when N^{+} is non-abelian. |N^{+} : L^{+}| = 2 by Theorem 1. Neither the dihedral group [12] nor the quaternion group [4] can be the additive group of a local near-ring. Since |N^{+}| = 8 in the case (6), it is impossible. Observe in the case (5) that (1 + L)^{∗} has a unique element of order 2 and it acts trivially on L^{+}. Therefore, for every l in L, l · (−1) = l and then −1 + l + 1 = −l by Lemma 2. But −1 + 2 · 1 + 1 = 2 · 1. Thus, 4 · 1 = 0. Consequently, 4 · L = 0 and |N| ≤ 8. No such near-ring exists by the same reason as above.

Now assume that N^{+} is abelian. If (L, +, ×) is given by (5), arguments as in 1 can be used. Suppose that (L, +, ×) is defined by (6). Denote by

C_{L^{+}}((1 + L)^{∗}) = {l ∈ L | l · (1 + x) = l for all x from L}

the centralizer of L^{+} with respect to the action of (1 + L)^{∗} by the right multiplication. It follows from (6) that C_{L^{+}}((1 + L)^{∗}) = ⟨u_{1} + u_{2}⟩. Thus N · (u_{1} + u_{2}) ≤ C_{L^{+}}((1 + L)^{∗}). Consequently,

(7)  r · (u_{1} + u_{2}) = u_{1} + u_{2} for r ∈ N^{∗},

and

(8)  L · (u_{1} + u_{2}) = 0

by Lemma 7. Using Lemma 7 again, u_{1}^{2} ≠ u_{1}. Suppose u_{1}^{2} = u_{2}. Note that u_{1} · (1 + u_{1}) = u_{1} + u_{1} × u_{1} = u_{2}, and u_{2} · (1 + u_{1}) = u_{2} + u_{2} × u_{1} = u_{1} according to (6). Multiplying the previous equation by 1 + u_{1}, obtain u_{1} · u_{2} = u_{1}. Then u_{1}^{3} = u_{1} · u_{1}^{2} = u_{1} · u_{2} = u_{1}. This contradicts Lemma 7. Consequently, u_{1}^{2} = 0 or u_{1}^{2} = u_{1} +u_{2}. In any case, u_{1}^{2} ∈ C_{L^{+}}((1+L)^{∗}), and u_{1} ·u_{2} = u_{1} ·u_{1} ·(1+u_{1}) = u_{1}^{2}. Similarly, u_{2} ·u_{1} = u_{2}^{2}. Let u_{1}^{2} = u_{1} +u_{2} and u_{2}^{2} = 0. Then u_{2} · u_{1} = u_{2}^{2} = 0, 0 = u_{1}^{3} = (u_{1} + u_{2}) · u_{1} = u_{1} + u_{2}. We have a contradiction. Therefore, L can have only two different multiplication tables:

(9)  u_{1}^{2} = u_{1} · u_{2} = u_{2}^{2} = u_{2} · u_{1} = u_{1} · (u_{1} + u_{2}) = u_{2} · (u_{1} + u_{2}) = 0
(10) u_{1}^{2} = u_{1} · u_{2} = u_{2}^{2} = u_{2} · u_{1} = u_{1} + u_{2}, u_{1} · (u_{1} + u_{2}) = u_{2} · (u_{1} + u_{2}) = 0

Using the homomorphism U_{l}, one obtains that |N^{+} : L^{+}| ≤ 4 and L is a null ring if |N^{+} : L^{+}| = 4.

Suppose that |N^{+} : L^{+}| = 4. There are elements r_{1} and r_{2} in N^{∗} such that r_{1} + r_{2} lies in N^{∗}. Consider the action of N^{∗} on L defined by the rule l → n · l for l from L and n from N^{∗}. Because L is a null ring, (1 + L)^{∗} centralizes L. Moreover, N^{∗}/(1 + L)^{∗} centralizes u_{1} + u_{2} according to (7). Since N^{∗}/(1 + L)^{∗} has an odd order, N^{∗} centralizes L. In particular, we have u_{1} = (r_{1} + r_{2}) · u_{1} = r_{1} · u_{1} + r_{2} · u_{1} = 2u_{1} = 0. This is a contradiction. Thus, the case |N^{+} : L^{+}| = 4 is impossible.

Finally, assume that |N^{+} : L^{+}| = 2. Note that the operation · given by the multiplication table (10) coincides with the operation × defined in (6). Consequently, we obtain rings in the case (10). According to [8], these are the rings Z_{2}[x]/ ⟨x^{3}⟩, Z_{4}[x]/ ⟨2x, x^{2} − 2⟩. Note that this fact can be checked directly analyzing the obtained equations.

Let the equations (9) hold, i.e., L has zero multiplication. Because 2 · (1 + x) = 2 + 2 · x = 2 for any x from L, 2 ∈ C_{L^{+}}((1 + L)^{∗}). There are two cases: 2 = u_{1} + u_{2}, 2 = 0. Combining (6) and (9), we obtain the near-ring NRC_{4,2} in the first case and the near-ring NRC_{2,2,2} in the second case. The multiplicative groups of these near-rings are isomorphic to the quasi-regular group of the ring (6) which is isomorphic to C_{4}.

□

4. Conclusion

The method of reduction to a radical rings has proved to be fruitful for the study of local near-rings with an abelian group of units. Though the main tool of this paper (Proposition 1) stipulates that the group of units is commutative, the principal idea of the proposition can be applied to the non-commutative cases as well. Let N be a local near-ring and L be the subgroup of elements without left inverse. Instead of radical ring, one should consider the triply factorized group G = AB = AM = BM where A and B are isomorphic to (1+L)^{∗}, M is isomorphic to (1 + L)^{+}, and the action is given by multiplication of L^{+} on the right side in N. Several structure theorems for the group G with given conditions on the factors are known [1]. Hence, some results of this paper can be generalized to other classes of groups. In particular, the case when the group of units is nilpotent can be explored.

In conclusion, the author wishes to express his thanks to Professor Ya. P. Sysak for fruitful discussions and useful remarks concerning this paper.

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Department of Mathematics, The Ohio State University, Columbus OH 43210

E-mail address: gorodnik@math.ohio-state.edu