Distribution and Generalized Centre in Planar Nearrings

Nearrings are the nonlinear generalization of rings, having only one distributive law. Planar nearrings are a special class of nearrings that generalize nearfields, themselves a generalization of fields, which play an important role in the structural theory of nearrings[14, 20], as well as having important geometric and combinatorial properties[6, 10, 15].

In this paper we look at the distributive elements of a planar nearring, in some sense the ring-like elements of the planar nearring. This extends Aichinger’s work on planar rings [2]. We find that the general structure of planar nearrings with nontrivial distributive elements generalizes the structure of planar rings. We use this information to then investigate the generalized center of a planar nearrings, building on Farag, Cannon, Kabza and Aichinger’s work [3, 7]. This was the original motivation for this work and can be found in §6. As the results there indicate, it was necessary to obtain a good understanding of the distributive elements of a planar nearring, which we undertake in §4 and find some small applications of in §5. The structure implied by nontrivial distributive elements forces several special forms of planar nearrings, which we introduce in §3 and use repeatedly throughout.

In the next section we introduce the necessary background about nearrings and planar nearrings.

A (right) nearring $(N,+,*)$ is an algebra such that

• •

  $(N,+)$ is a group with identity $0$

• •

  $(N,*)$ is a semigroup

• •

  for all $a,b,c\in N$, $(a+b)*c=a*c+b*c$ (right distributivity)

While it is readily seen that $0*a=0$ for all $a\in N$, it is not necessary that $a*0=0$ for all $a$. If this is the case, the nearring is called zero symmetric.

A subset of $N$ is a subnearring if it is closed as an additive group and a multiplicative semigroup. A subnearring $I\subseteq N$ is a right ideal if $I$ is a normal additive subgroup and for all $i\in I$, $n\in N$, $i*n\in I$. A subnearring $I\subseteq N$ is a left ideal if $I$ is a normal additive subgroup and for all $i\in I$, $n,m\in N$, $n*m-n*(m+i)\in I$. A subnearring that is a left and right ideal is an ideal. As we expect, ideals correspond to the kernels of nearring homomorphisms.

We write $N^{*}$ for $N\setminus\{0\}$. If $(N^{*},*)$ is a group, then $(N,+,*)$ is a nearfield, generalizing fields by having a single distributive law. Nearfields play an important role in geometries, nearring and nearfield theory are well discussed in [10, 11, 17, 19]. Nearfield were first described by Dickson using a process that is now called a Dickson nearfield. All but 7 finite nearfields are Dickson.

The set $D(N)=\{n\in N|n(a+b)=na+nb\,\forall a,b\in N\}$ of distributive elements of $N$ is the core investigation of this paper.

Elements $a,b\in N$ are called equivalent multipliers if $x*a=x*b$ for all $x\in N$. We write $a\cong b$, an equivalence relation. A nearring $(N,+,*)$ is called planar if

• •

  The equivalent multiplier equivalence relation has at least 3 classes

• •

  For every $a,b,c\in N$, $a\not\cong b$, the equation $x*a=x*b+c$ has a unique solution.

Every field other than $\mathbb{Z}_{2}$ is a planar nearring. A finite nearfield (except $\mathbb{Z}_{2}$) is always planar, there are known to be nonplanar infinite nearfields [19, page 46]. Planar nearrings are zero symmetric.

Planar nearrings can be described by fixed point free automorphism groups of groups. Let $(N,+)$ be a group. Then we say that some nonidentity $\phi\in Aut(N)$ is fixed point free if for all $n\in N$, $n\phi=n$ iff $n=0$. A group of automorphisms is fixed point free if all nonidentity elements are fixed point free. Now let $\Phi\leq Aut(N)$ be a group of fixed point free automorphisms of $N$ acting from the right such that $-id+\phi$ is bijective for all non identity $\phi\in\Phi$. We write $a\Phi$ for the orbit containing $a$. Let $R\subseteq N$ be a set of orbit representatives, $M\subseteq R$ a set. Every element $a\in N$ can be written uniquely as $r_{a}\phi_{a}$ for some $r_{a}\in R$, $\phi_{a}\in\Phi$. We define a multiplication by:

Then $(N,+,*)$ is a planar (right) nearring. All planar nearrings can be so derived[10].

Let $a\Phi^{*}=a\Phi\cup\{0\}$. We call $M$ the set of zero multipliers. If $a\in M$ then we call $a\Phi$ a zero multiplier orbit, and we will call all elements of this orbit zero multipliers. Then $n\in N^{*}$ is a zero multiplier iff $n\in M\Phi^{*}$ iff $a*n=0$ for all $a\in N$. Note that $a*b=0$ iff $a=0$, $b=0$ or $r_{b}\in M$. In particular, $n*b=0$ for all $n$ iff $b=0$ or $r_{b}\in M$. The elements $r\in R\setminus M$ are right identities, $x*r=x$ for all $n\in N$. A planar nearring has a left identity iff it has an identity iff it has exactly one nontrivial $\Phi$ orbit iff it is a planar nearfield. We use $Z(\Phi)$ to denote the centre of the group $\Phi$ and remark that we will use the British spelling throughout this paper. The distributive elements of a nearfield are called the kern of the nearfield and contains the multiplicative centre.

There are related nearrings. The trivial nearring on any group $(N,+)$ with multiplication $a*b=0$ would correspond to $\cong$ having one equivalence class. The Malone trivial nearrings [8] correspond to $\Phi$ having order 1, so $R=N\setminus\{0\}$, $a*b=0$ if $b\in M$ and $a*b=a$ otherwise. It is interesting to note that the complemented Malone nearrings in [8] are a generalization of planar nearrings with $\Phi$ of order 2, allowing fixed points (i.e. elements of additive order 2) that are zero multipliers.

In the next section we will look at two special constructions for planar nearrings. Then in the following section we look at the distributive elements of a planar nearring. With that knowledge, we will determine the generalized centre of a planar nearring.

In this section we construct two families of example planar nearrings, which will prove to be useful in the rest of the paper.

Let $V$ be a vector space over a division ring $D$ of order at least 3 and $\phi:V\rightarrow D$ a vector space epimorphism which acts from the right. Define a multiplication $*:V\times V\rightarrow V$ by $a*b=a(b\phi)$. By [2, theorem 4.1] and [21, theorem 5.2.1] this is a planar ring and all planar rings have this form. Using the terminology above, $\Phi$ is isomorphic to the nonzero elements of $D$ under multiplication, $R\setminus M=\phi^{-1}(1)$ and the elements of $M$ can be chosen arbitrarily from the orbits that lie completely within $\ker\phi$. Let $v_{1}\in R$ be arbitrary. If $V$ is a finite dimensional vector space, we can choose a new basis $v_{1},\dots,v_{n}\in V$ such that $v_{2},\dots,v_{n}\in\ker\phi$ and for all $x=(x_{1},\dots,x_{n})\in V$, $x\phi=x_{1}$.

We can generalize this construction to nearvector spaces¹¹1We note in passing the existance of another, similar, definition of a nearvector space used by Karzel and colleagues[12, 13] in which the right nearfield scalars operate from the left, giving significantly different properties.. We begin with a brief overview and some definitions of nearvector spaces. See [5] for further details.

We sometimes refer to $V$ as a nearvector space over $A$. We write $Q(V)$ for the quasi-kernel of $V$. The elements of $V$ are called vectors and the members of $A$ scalars and it turns out that $A$ is a nearfield. The action of $A$ on $V$ is called scalar multiplication. Note that $(V,+)$ is an abelian group. Also, the dimension of the nearvector space, $\dim(V)$, is uniquely determined by the cardinality of an independent generating set for $Q(V)$.

In [18, theorem 3.4] we find a characterization of finite dimensional nearvector spaces, see also [5, theorem 4.6].

In [5, 4.13 ff] we find the following. A nearvector space is regular if all the $\psi_{i}$ are identical (up to nearfield automorphisms). Every nearvector space has a unique maximal decomposition $V=V_{1}\oplus V_{2}\oplus\dots$ into regular sub-nearvector spaces $V_{i}$. Then for all nonzero $u\in Q(V)$, there is precisely one $i$ auch that $u\in V_{i}$. Note that a regular vector space over a field is a vector space.

We can construct a planar nearring from a near vector space along the lines used above for vector spaces. Let $V$ be a nearvector space over a nearfield $F$ of order at least 3. Let $\phi:V\rightarrow F$ be a nearvector space epimorphism and define $a*b=a(b\phi)$. The right identities $R\setminus M$ are $\phi^{-1}(1)$ and the representatives in $M$ can be chosen arbitrarily.

It is worth noting in passing that nearvector spaces and the homogeneous mappings of them to themselves are closely related to questions about nearring matrices over the associated nearfield. Thus we hope that future work here could shed light on the question raised in the final section of [14] as to the inverses of units in matrix nearrings over planar nearfields.

In this section we investigate the distributive elements of a planar nearring. Some examples are well known. A finite field is a planar nearfield and thus a planar nearring, with all elements being distributive. In [2] (see §3 above) the structure of planar rings is completely determined, so we know what happens when $D(N)=N$. The distributive elements of a nearfield are called the kern of the nearfield.

Using Sonata [1] we found all planar nearrings up to order 15 with nontrivial $D(N)$.

1. 1.

   The fields of order 3,4,5,7,8,9,11,13.

2. 2.

   The proper nearfield of order 9 with kern of order 3.

3. 3.

   The planar ring of order 9.

4. 4.

   An example of order 9. The additive group is $\mathbb{Z}_{9}$, $\Phi=\{1,-1\}$, $R=\{2,3,5,8\}$, $M=\{3\}$. The distributive elements are the zero multipliers $\{0,3,6\}$.

5. 5.

   An example of order 15, $\Phi$ of order 2 with generator $g$ acting on $\mathbb{Z}_{3}\times\mathbb{Z}_{5}$ as $(x,y)*g=(-x,-y)$, the orbit $\{0\}\times\mathbb{Z}_{5}$ zero multipliers. The distributive elements are $\mathbb{Z}_{3}\times\{0\}$.

We see that 1-3 can be readily explained, but not 4 or 5. One of the goals of this paper is to understand all these examples in terms of general classes.

We first determine some properties of the orbits that contain distributive elements.

Similarly we know that, even in the case that there are distributive elements that are zero multipliers, additive closure of the orbit $d\Phi^{*}$ allows us to define a multiplication $d\phi_{1}\circ d\phi_{2}=d(\phi_{1}\phi_{2})$ that gives us $(d\Phi^{*},+,\circ)$ a planar nearfield.

Thus we know a lot more about the forms of $\Phi$ that can emerge, as not all fixed point free automorphism groups arise as the multiplicative group of a nearfield.

Note that, in general, the mapping $\rho$ is not a nearring homomorphism, unless $\phi_{d}$ is the identity. This can only be guaranteed to be the case when $D(N)$ contains non zero multipliers, by lemma 4.3.

This implies that $(N,+)$ is an extension of $(\rho(N),+)$ by $(K,+)$. By lemma 4.5 and the comments afterwards, we know that $(\rho(N),+)$ is the additive group of a nearfield, thus abelian and, in the finite case, elementary abelian. When all distributive elements are zero multipliers, we do not necessarily have that the extension splits. If we have a non zero multiplier distributive element, then we get a clear result.

We see that the example on additive group $(\mathbb{Z}_{9},+)$ falls outside this theorem. The distributive elements lie within the zero multipliers and the additive group is not a semidirect product of the zero multipliers with anything.

We can create a family of examples of planar nearrings with $D(N)$ lying within the zero multipliers, based upon the example on page 49 of [10]. These examples do not split.

We know (e.g. [16]) that the additive group of a finite planar nearring is nilpotent and thus a direct sum of $p$-groups. Thus a finite planar nearring is a finite direct sum of planar nearrings of prime power order. Thus by lemma 4.5 at most one of these summands has a non zero multiplier distributive element. If one summand has such an element, then lemma 4.9 indicates that we have a trivial multiplication of all summands other than the one with a non zero multiplier distributive element. We have shown the following.

In the example of order 15, we have the $\mathbb{Z}_{3}$ as the nearfield with automorphism group of order 2 and $\mathbb{Z}_{5}$ having the same automorphism group generated by $(1\,4)(2\,3)$ acting on it. Then $N=\mathbb{Z}_{3}\times\mathbb{Z}_{5}$ with $\{0\}\times\mathbb{Z}_{5}$ forming the zero multipliers in the nearring.

Thus we have shown all of our small examples of planar nearrings with nontrivial distributive elements fall into larger classes of examples.

In this section we look at the way that these results can be contextualized in relation to other similar results.

One of the strong applications of planar nearrings is in the construction of BIBDs. The blocks of the BIBD are $\{a\Phi^{*}+b|0\neq a\in N,b\in N\}$ and the basic blocks are $\{a\Phi^{*}|0\neq a\in N\}$. The following result shows that the construction used in [9] is the only way that all basic blocks can be subgroups.

We note in passing that if a planar nearring has non trivial distributive elements, then we know that the corresponding orbits give additively closed basic blocks. Thus by [10, Thm 7.17] we will obtain a statistical, but not a geometric BIBD.

We also obtain Aichinger’s result as a corollary.

Now that we know some things about the distributive elements of a planar nearring, we can say some things about the generalized centre.

Let $(N,+,*)$ be a 0-symmetric nearring. The generalized centre of $N$ is $GC(N)=\{n\in N|nd=dn\;\forall d\in D(N)\}$ [3, 7].

The generalized centre was introduced because the centre of a nearring is not always well behaved. For instance, it is not always a subnearring, while the generalized centre is. If $N$ is a ring, then $GC(N)$ is the usual centre of $N$.

The generalizes centre of a planar nearfield $F$ is the set of elements that commute with the kern. In the finite case, the kern is the multiplicative centre and thus $GC(F)=F$. In the infinite case, when the kern is distinct from the multiplicative centre, we know only that $GC(F)$ contains the multiplicative centre.

We can break this down depending upon the properties of the additive group.

If the additive group is abelian, then many interesting and strange things can happen with skew fields. However in the finite case we know more.

In this paper we have investigated the distributive elements in a planar nearring. We have been able to show that if there are nontrivial distributive elements then the additive group is an extension of an abelian subgroup by the zero multipliers. This additive group is the additive group of a nearfield, so elementary abelian in the finite case. If the distributive elements include non zero multipliers, then the extension splits and we obtain a clear structure.

As a result, we are able to re-prove Aichinger’s theorem on planar rings as a corollary, as well as Clay’s results on BIBDs with additively closed basic blocks. It is unclear whether lemma 5.1 can be extended to the infinite case. Applying these results to the question of the generalized centre, we are able to obtain a clear set of cases and to describe the generalized centre.

It would be valuable to know what sort of other examples can occur with the distributive elements all lying within the zero multipliers, in order to complete the classification of structures.

It would be of value to calculate $D(N)$ explicitly in theorem 1. It is easy to see that $D(N)$ is a direct product of some subset $E\subseteq K$ and the kernel of $F$. The question is how to calculate which parts of $K$ have $a(\phi_{b+c})=a\phi_{b}+a\phi_{c}$ where the first addition is in $F$ while the second is in $K$. Note that the orbits will be additively closed, giving us nearfields. This might be another nearvector space construction.

As we have been able to determine the generalized centre of planar nearrings, we can now look forward to describing the generalized centre of more complex classes of nearrings.

Research supported by SFB Project F5004 of the Austrian Science Foundation, FWF. I would like to thank my colleagues Günter Pilz and Wen-Fong Ke for some insightful questions in the early development of this paper.