Abstracts of Nearrings and Nearfields Conference on
Nearrings and Nearfields

27 July 2003 - 3 August 2003

Abstracts

In case of multiple authors, the speaker is indicated by a star*.
The list is sorted by the last name of the speaker.

Bounds on the size of N-groups that lead to decidability results

Erhard Aichinger, Linz, Austria

Given a finite zero-symmetric near-ring with identity N, we ask whether N has any of the following properties:

1. There is a group G such that N is isomorphic to the inner automorphism near-ring I(G).
2. There is a finite group G such that N is isomorphic to the near-ring I(G).
3. N is compatible, i.e., it has a faithful compatible N-group.

In particular, we investigate whether there is an algorithm that tells whether a given N has any of these properties. The fact that such an algorithm exists is based on the following theorem:
Theorem. Let N be a finite near-ring, and let m : = |N|. If there is a group G such that N is isomorphic to I(G), then there is a group H with |H| £ m2m+1 such that N is isomorphic to I(H).
Using a similar result, we prove that is decidable whether a given near-ring is compatible in the sense of S.D.Scott.

Recently, we have proved that the variety of near-rings is generated by its finite members. We explain which information this gives on the decidability of certain word-problems for near-rings.

On prime near-rings with semiderivations

Nurcan Argaç, Izmir, Turkey

Throughout this paper N always stands for a right near-ring. An additive map d:N® N is a derivation if d(xy) = xd(y)+d(x)y for all x,y Î N - or equivalently ( cf. [6] ) that d(xy) = d(x)y+ xd(y) for all x,y Î N . The study of derivations of near-rings was initiated by H.E. Bell and G. Mason in 1987 [3], but up to now only a few papers on this subject in near-rings were published ( see [1], [2] , [4] and [5] ). Now we introduce the notion of semiderivation of a near-ring N as follows.

Definition . An additive mapping d:N® N is a semiderivation if there exists a function a: N® N such that
 d(xy) = d(x)a(y)+xd(y) = d(x)y+a(x)d(y)
for all x,y Î N.

Theorem 1 Let N be a semiprime right near-ring and I be a subset of N such that 0 Î I and IN Í I. Let d be a semiderivation on N with associated function a such that a(I) = I and a(xy) = a(x)a(y) for all x, y Î I.
(i) If d acts as a homomorphism on I, then d(I) = {0}.
(ii) If d acts as an anti-homomorphism on I and a(0) = 0 , then d(I) = {0}.

Theorem 2 Let N be a prime near-ring , I a nonzero semigroup ideal of N and d a nonzero semiderivation of N. If d(x +y -x-y) = 0 for all x,y Î I, then ( N,+ ) is abelian.

## References

[1]
Argaç, N.: On prime and semiprime near-rings with derivations. Internat. J. Math. and Math. Sci. 20 (4) (1997), 737-740.
[2]
Beidar, K.I, Fong, Y. and Wang, X.K.: Posner and Herstein theorems for derivations of 3-prime near-rings. Comm. Algebra, 24(5) (1996), 1581-1589.
[3]
Bell, H. E. and Mason, G.: On derivations in near-rings, Near-rings and Near-fields, North-Holland Mathematics Studies 137 (1987), 31-35.
[4]
Bell, H.E. and Argaç, N.: Derivations, products of derivations, and commutativity in near-rings. Algebra Colloq. 81(8) (2001), 399-407.
[5]
Hongan, M.: On near-rings with derivation. Math. J. Okayama Univ. 32 (1990), 89-92.
[6]
Wang, X.K.: Derivations in prime near-rings. Proc. Amer. Math. Soc. 121 (2), 1994, 361-366.

Freiman Nearrings

H. Bell, St. Catharines, Ontario

A nearring is called a Freiman nearring if the square of each 2-subset contains at most 3 distinct elements. We discuss sufficient conditions for a Freiman nearring to be additively or multiplicatively commutative.

PBIBDs from weakly divisible nearrings and related codes

Anna Benini, Brescia

In previous papers a class of finite nearrings called "weakly divisible" (wd-nearrings) is defined, studied [1] and constructed [2] and a family of partially balanced incomplete block designs (PBIBDs) is derived [3]. Now the design parameters and the design incidence matrix are investigated in order to study the related codes: row code, column code and their linear hulls.

1. Benini, A. and Pellegrini S., Weakly Divisible Nearrings, Discrete Math., 208/209 (1999), 49-59.
2. Benini, A. and Morini F., On the construction of a class of weakly divisible nearrings, Riv. Mat. Univ. Parma, (6) 1 (1998), 103-111.
3. Benini, A. and Morini F., Partially Balanced Incomplete Block Designs from Weakly Divisible Nearrings, submitted.

Radical parallelism: A geometric interpretation of the Jacobson radical of a ring

Andrea Blunck, Hamburg

We introduce a relation called radical parallelism on the projective line over a ring R in a purely synthetic way, using only the notion of distant points. It turns out that this relation is an equivalence relation that is closely related to the Jacobson radical rad(R). In particular, rad(R) can be identified with a radical parallel class in the affine trace of a fixed point at infinity.
(Joint work with Hans Havlicek, Vienna University of Technology.)

Primeness and Radicals in Near-rings of Continuous Functions

G.L. Booth, University of Port Elizabeth, Nelson Mandela Bay, South Africa

Let (G,+) be a group.  The set of all zero-preserving self-maps of G is denoted M0(G), i.e. M0(G): = {a:G® G | a(0) = 0}.

Let (G,+) be a topological group.  The set of all continuous zero-preserving self-maps of G is denoted N0(G), i.e. N0(G): = {a:G® G | a is continuous and a(0) = 0}.  Note that if G has the the discrete topology, M0(G) = N0(G).

All topological groups in this talk will be T0, and hence completely regular. It is clear that M0(G) and N0(G) are (right) zerosymmetric near-rings with respect to the operations pointwise addition and composition of functions.  Various generalizations of primeness exist in near-ring literature.  We will be considring the classical definition, as well as 3-primeness (Groenewald, 1989) and equiprimeness (Booth, Groenewald and Veldsman, 1991).  We will also consider two generalisations of stronlgly prime, the first of which is known by the name, and the second is called strongly equiprime.  The prime and equiprime radicals of N will be denoted P(N) and Pe(N), respectively.

Veldsman (1992) has noted that M0(G) is always equiprime.  The next two results shed some light on the situation for N0(G).

Definition 1 A topological space X is called 0-dimensional if the topology on X has a base consisting of clopen (i.e. both open and closed) sets.

Proposition 2 Let G be a topological group with more than one element which is either 0-dimensional or arcwise connected.  Then N0(G) is equiprime.

Proposition 3 Let G be a disconnected topological group, with open components which are arcwise connected and which contain more than one element.   Let H be the component of G which contains 0, I: = {a Î N0(G) | a(G) Í H} and J: = {a Î N0(G) | a(H) = 0}.  Then P(N0(G)) = Pe(N0(G)) = IÇJ.

Remark 4 In this case P(N) ¹ 0, so N0(G) is not semiprime.

Proposition 5 Let G be a 0-dimensional topological group with more than one element.  Then

1. N0(G) is strongly prime if and only if the topology on G is discrete.

2. N0(G) is strongly equiprime if and only if G is finite.

Proposition 6 Let on G be an arcwise connected topological group whichhas a base B consisting of arcwise connected open sets (i.e. G is locally arcwise connected).  Then N0(G) is not strongly prime (and hence not strongly equiprime).

Remark 7 It is not known whether there exists a non-discrete topological group G such that N0(G) is strongly prime.

Let X and G  be a topological space and a topological group respectively, and let q:G® X be a continuous map.  The sandwich near-ring N0(G,X,q) is the set {a:X® G | a is continuous and aq(0) = 0}.  Addition is pointwise and multiplication is defined by a·b: = aqb.  If the topologies on X and G are discrete we denote the near-ring by M0(G,X,q).  In this talk we will assume that both G and X have more than one element.

Proposition 8 (Veldsman, 1992) M0(G,X,q) is equiprime if  and only if q is a bijection.  In this case M0(G,X,q) @ M0(G).

Proposition 9 Suppose that X is a 0-dimensional, T0 space and G is a 0-dimensional, T0 topological group.  Then N0(G,X,q) is equiprime if  and only if q is injective and c(q(G)) = G.

Proposition 10 Let X and G be a completely regular topological space and an arcwise connected topological group, respectively, and let g:G® X be a continuous, injective map.  Then the following are equivalent:

(a)       cl(g(G)) = X.

(b)       N0(G,X,g) is equiprime.

(c)       N0(G,X,g) is 3-semiprime.

Difference Methods and Ferrero Pairs

Tim Boykett* and Peter Mayr

There are several constructions of designs from planar nearrings. In this paper, we present a generalised theory for constructing BIB-designs from a group G and a group F of fixed point free automorphisms of this group. Using this theory we can unify several previous constructions of BIB-designs. In particular we can apply Sun's results on lines and segments in planer nearrings to this more general situation.

Generalized Centers of Nearrings

G. Alan Cannon, Hammond, LA, U. S. A.

Let N be a right nearring. Denote by C(N) the multiplicative center of N and by Nd the set of left distributive elements of N. In general, C(N) need not be closed under the addition of N. However, the generalized center of N, GC(N) = {a Î N  | and = nd a for all nd Î Nd}, is always a subnearring of N containing C(N). We initiate a study of the structure of GC(N) as well as when GC(N) equals C(N) for certain classes of nearrings.

Finite affine complete groups

Jürgen Ecker, Linz

A mapping on a group G which respects all congruence relations of G is called a compatible function. The compatible functions on a group G form a subnearring C(G) of the nearring M(G) of all mappings on G. The inner automorphism nearring I(G) is a prominent subnearrings of C(G). A group G is called 1-affine complete if I(G) = C0(G). In my talk I describe a few methods needed to find the 1-affine complete Frobenius groups and the 1-affine complete generalised dihedral groups. I show which quotients of a group inherit 1-affine completeness, and I present an example of a 1-affine complete group G such that the direct product G×G is not 1-affine complete.

Jacobson Near-rings

Nico Groenewald

In this talk we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that most cases the class of all near-rings of this type are special radical classes. Subsequently we investigate the relationship between each Jacobson-type of near-ring and the corresponding matrix near-ring.

On class pairs and radicals in near-rings

Lungisile Godloza

In this talk we inestigate the properties of classes near-rings constructed usind a class pair (M1:M2) of near-rings. We establish the relationship between a class pair (M1:M2) and the radical pair (r1:r2) constructed using the preradicals r1 and r2 associated with the classes M1 and M2 respectively. Conditions under which (r1:r2) = (Sr2:Sr1), where Sri is the semisimple class of ri, i = 1,2, and is a Kurosh-Amitsur radical class are established.

Classification of type-0 R-groups and the nilpotence level of the s-radical

J.F.T. Hartney, Johannesburg, South Africa

For d.g. near-rings R, Laxton and Machin [1] constructed an antiradical, which they called the critical ideal by using a classification of R-groups of type-0 into two classes H1 and H2. Two assumptions were made with regard to H1 and H2. In addition, H1 was assumed to be the maximal class for which these assumptions were valid. The R-groups of type-0 were all subfactors of a faithful R-group W and the critical ideal, Crit(R), was defined as the annihilator of the class H2. Crit(R) is independent of W as long as H1 is a maximal class.

Our classification is slightly different and extends to arbitrary zero-symmetric near-rings R. For near-rings with DCCL we define C(W) as the annihilator of one of the classes obtained from a faithful R-group W. For W = R, C(R) = Soi(R), the socle-ideal of R. For near-rings with DCCR Crit(R) = C(W) = Soi(R) for any faithful R-group W. We then investigate how the classification is affected as we move along a nil-rigid series for R. This enables one to view the nilpotence level (cf.[2]) of the s-radical Js(R) via faithful R-groups.

[1] LAXTON, R.R. and MACHIN, A.W. On the decomposition of near-rings. Abh. Math. Sem. Univ. Hamburg 38 (1972), 221-230.
[2] HARTNEY J.F.T. On the decomposition of the s-radical of a near-ring. Proc. Edinb. Math. Soc. 33 (1990) 11-22.

The Minimal and Maximal Right Ideals in the Nearring of Polynomials

Lucyna Kabza, Hammond, LA, USA

Let R be a commutative ring with identity. Then (R[x],+,°) is a nearring of polynomials with the usual addition and function composition. The maximal ideals of this nearring were studied by J. R. Clay and D. K. Doi (with R = F, F an arbitrary field) and by H. Kautschitsch, who determined all maximal ideals over an arbitrary commutative ring R with identity with more than two elements. In this paper the minimal and maximal right ideals of R[x] are studied and some necessary properties of these ideals are established. Moreover, some maximal right ideals of Z2[x] are explicitly given.

Ideals in the nearring of formal power series over local rings

Herrmann Kautschitsch

In this talk an overview about a special class of nearring ideals, which includes prime ideals, is given. It is also shown, how they are related to the ideal structure of the underlying coefficient ring and how they are connected to the ring ideals.

On Recent developments of planar nearrings

Wen-Fong Ke, Tainan

In this talk, I shall present some on-going researches on planar nearrings. For example, we have obtained a classification of isomorphic planar nearrings constructed from a fixed Ferrero pair, and the description of the automorphism group of some designs constructed. Some emphases will also be put on circular planar nearrings including the realization of the regular group of automorphisms, and the ``overlapping" of the graphs.

On Finite Goldie Dimension of Mn(N) Group Nn

Syam Prasad Kuncham, Manipal Institute of Technology, India

In this paper we studied the concepts: linearly independent elements and u-linearly independent elements in N-group G where N is a zero symmetric right near-ring. We proved that the Goldie dimension of the N-Group N is equal to that of the Mn(N)-group Nn where Mn(N) is the matrix near-ring.

J.F.T. Hartney and A.M. Matlala*, Johannesburg, South Africa

The socle ideal of R , Soi(R), is an anti-radical in the sense that it annihilates at least one Jacobson radical, in particular J[1/2](R) . We show that Soi( Mn (R) ) Í ( Soi(R) )* , where R is a zero-symmetric near-ring with identity and Mn (R) the matrix near-ring associated with R . It is not known whether the strict inclusion holds. The importance of the socle-ideal, for our purposes, is in investigating the nilpotency of the s-radical, Js (R) , since Soi(R) is a non-nilpotent part of a near-ring. It is not known if (J0 (R))+ Í J0 ( Mn (R) ) for general near-rings, as pointed out by Meldrum and Meyer [5]. We prove that (Js (R))+ Í Js ( Mn (R) ) for near-rings satisfying DCCR. In fact, the example given in [5] is a special case of our result as the given near-ring is such that Js (R) = J0 (R) . By using a special nil-rigid series, as defined by Scott [6], a unique minimal ideal A such that Js (R) / A is non-zero nilpotent was defined under suitable chain conditions by Hartney [3]. We call this ideal A the s-socle of R . For finite near-rings R we show that A+ Í A , where A is the s-socle of Mn (R) .

Completeness for concrete near-rings

Dragan Masulovi\'c, Novi Sad (Serbia and Montenegro)

In this talk a completeness criterion for near-rings over a finite group is derived using techniques from clone theory. The relationship between near-rings and clones containing the group operations of the underlying group shows that the unary parts of such clones correspond precisely to near-rings containing the identity function. Rosenberg's characterization of maximal clones is then applied to describe maximal near-rings containing the identity map, while maximal near-rings not containing the identity are described using typical near-ring methods. This finally provides us with a completeness criterion. We apply this criterion to show that if the order of G is large then with high probability the set containing a single bijection is complete.

This is a joint work with E. Aichinger (Linz), R. Pöschel (Dresden) and J. S. Wilson (Birmingham).

Near-Rings of Mappings

Carl J. Maxson

In this talk we discuss some old and some new avenues of investigation related to near-rings of mappings.

Polynomial functions on linear groups

Peter Mayr, Linz, Austria

For the non-solvable linear, unitary, symplectic, and orthogonal groups over finite vector-spaces and related groups G, we consider the following problems: Determine the size of the inner automorphism near-ring I(G). Do we have I(G) = A(G)? Do we have I(G) = E(G)?

Our results are based on a description (jointly done with E. Aichinger) of the unary polynomial functions on those non-abelian finite groups G that satisfy the following conditions: G¢ = G¢¢, G/Z(G) is centerless, and there is no normal subgroup N of G with G¢ÇZ(G) < N < G¢.

Some recent developments in group near-rings

J. D. P. Meldrum, Edinburgh

Following the definition of matrix near-rings in 1986 by Meldrum and van der Walt, which has proved of interest, a definition of group near-rings was presented by Le Riche, Meldrum and van der Walt in 1989 using similar ideas. Here Group Near-rings are discussed emphasizing the parallels between them and matrix near-rings. The use of these ideas to develop a theory of polynomial near-rings is presented as developed by Bagley and Farag.

The main part is an account of work with J. H. Meyer on ideals in group near-rings. As with matrix near-rings, the relation between the ideals in the underlying near-ring and those in the group near-ring is much more complicated than in the case of rings. Added to that the situation for group rings is itself more complicated than for matrix rings and this is reflected in the near-ring case. To each ideal in the underlying near-ring corresponds a lattice of ideals in the group near-ring and some information about these lattices is given. There are also ideals in the group near-ring, the exceptional ideals, which do not fit into these lattices.

One of the most interesting of these exceptional ideals is the augmentation ideal. This is characterised and in all but a few exceptional cases, necessary and sufficient conditions are obtained for the augmentation ideal to be nilpotent. These involve generalised distributivity, the arithmetic structure of the group and the characteristic of the near-ring. The proofs involve a variety of techniques.

Modules over group near-rings

JH Meyer, Bloemfontein, South Africa

The notion of a module over a group near-ring is discussed. Interesting results concerning simplicity are given. This leads to unexpected interplay between the Jacobson radicals of the near-ring R and the group near-ring R[G].

Seminearrings of bivariate polynomials

Kent M. Neuerburg, Hammond, Lousiana, USA

The idea of the composed product of univariate polynomials may be generalized to bivariate polynomials. Using these root-based compositions, we define operations on bivariate polynomials analogous to addition and composition of univariate polynomials. Using these operations we investigate the seminearring of bivariate polynomials looking at its properties and internal algebraic structures.

Automorphism groups emitting local endomorphism nearrings. II

Gary L. Peterson, Harrisonburg, Virginia USA

A (left) nearring R with identity is local if the set of elements of R that do not have right inverses is a right R-subgroup of R. The study of local endomorphism nearrings of groups was initiated by Carter Lyons and the author [Proc. Edinburgh Math. Soc., 31(1988), 409-414; MR 89m:16077]. Several other papers dealing with local endomorphism nearrings by the author appeared thereafter. The primary focus of this note is to extend the results of one of these papers of the same title [Proc. Amer. Math. Soc., 105(1989), 840-843; MR 89k:20051].

Loop-nearrings

Silvia Pianta, Brescia

A loop-nearring is a triple (L,+,·) where (L,+) is a loop, with neutral element 0, (L*,·) is a semigroup (L*: = L\{0}) and the left (right) distributive law holds.

This notion arises naturally from the study of semigroups of endomorphisms of loops or, more generally, considering the set M(L) of all mappings from L to L. The interest of this kind of structure lies not just in the possibility of extending a number of well-known results already valid for nearrings to the case where the addition is not associative, but also, and expecially, in the discovery of a wide range of choices for such generalization.

The beauty of it is that, at least for some extent, each different choice fits with some suitable interesting class of examples, coming from the families of A-loops, K-loops or Moufang loops which take most of their significance from the foundation of geometry and the theory of permutation groups.

A right radical for right d.g. near-rings

J.F.T. Hartney and D.S. Rusznyak*, Johannesburg, South Africa

We discuss a Jacobson-type radical, rJ0(R)), for right d.g. near-rings.  rJ0(R) is defined using annihilators of certain d.g. right R-groups, which are the equivalent of type-0 R-groups from left representation. We know for d.g. near-rings R satisfying the DCC for left R-subgroups of R that  rJ0(R) contains the left 0-radical J0(R). For such d.g. near-rings we have J0(R) Í  rJ0(R) Í J2(R). Since J0(R) Í Js(R) Í J2(R), our main focus is connections between Js(R) and  rJ0(R).

Linearly Independent Elements in N-groups with Finite Goldie Dimension

Bhavanari Satyanarayana* (Nagarjuna University, India) and Syam Prasad Kuncham (Manipal Institute of Technology, India)

The concepts ``linearly independent elements'' and ``u-linearly independent elements'' in N-group G where N is a near-ring, were introduced and studied. Few important results that exist in the theory of Vector Spaces were obtained for N-groups. An N-group G is said to have Finite Goldie Dimension if it contains no direct sum of infinite number of non-zero ideals. We proved that (i) if G, G1 are two N-groups, f : G® G1 is an isomorphism, and ui belongs to G (for i = 1 to n), then ui's are u-linearly independent in G if and only if f(ui)'s are u-linearly independent in G1; (ii) if G has finite Goldie dimension, then K is a complement ideal of G if and only if there exist u-linearly independent elements ui < n and ui's (for i = 1 to k) are u-linearly independent elements in G, then there exists uk+1,..., un in G such that ui's (for i = 1 to n) are u-linearly independent elements of G which spans G essentially.

A.M.S. Subject Classification: 16A55, 16A66, 16A76 Key Words: Goldie dimension, uniform ideal, complement, linearly independent elements, essentially spanning subset, u-linearly independent elements.

The Z-constrained Conjecture

S. D. Scott

The talk starts by introducing compatibility. It is asked if, there is some reasonable condition for faithful compatible N-groups to be unique. A fairly well known theorem is stated. It uses r-constraint. This result can be made very much stronger by using Z-constraint. That is what this talk is about (the Z-constrained conjecture). Z-constraint yields nine equivalent conditions. In attempting to prove the Z-constrained conjecture, faithful hulls arise. These tend to tell us what the real nature of the theorem must be. How this is proved is outlined. Fitting factors are involved. They are subdirect sums. Proving the uniqueness of their components is a big step toward establishing the Z-constrained conjecture. The other big step is proving that, in the Z-constrained case, subdirect sums are unique.

Centralizer near-rings, matrix near-rings and cyclic p-groups

Kirby C. Smith, College Station

If G is a finite group and A is a group of automorphisms of G, then it is known that the matrix near-ring \mathbbMm(MA(G); G) is a subnear-ring of the centralizer near-ring MA(Gm) for every m ³ 2. Conditions are known under which \mathbbMm(MA(G); G) is a proper subnear-ring of MA(Gm), and if A and G are abelian, then conditions are known which imply the equality \mathbbMm(MA(G); G) = MA(Gm). In this paper we characterize the groups A of automorphisms of a cyclic p-group G for which this equality holds. We also show that, for every group A of automorphisms of a cyclic p-group G, either all the nonzero orbits of G are of unique type or none of the orbits of G is of unique type if p is odd, and there is a third possibility if p = 2, namely precisely one of the orbits of G is of unique type.

Anwendungen der Computer Graphik in der Geometrie

Grozio Stanilov

Mit jedem Dreieck verbinde ich 6 merkwuerdige Punkte, so dass bekommt man zwei neue Dreiecke, die homethetisch zu dem vorgegebenen sind. Auch 6 Ellipsen werden eingefuehrt - 3 davon eingeschriebene und 3 umgeschriebene sind, 6 Hyperbeln begleiten das Dreieck und noch 6 Parabeln. Alle besondere Punkten und Geraden dieser Kurven zweiter Ordnung sind neue merkwuerdige Objekten in der Dreiecksgeoemtrie. Alle Objekten werden mittels Maple 8 visualisiert.

Cohomology of near rings and applications

Mirela Stefanescu, Ovidius University, Constantza, Romania

The cohomological methods in studying algebraic structures give the possibility of constructing new structures and to deduce some properties of them from properties of associated groups. This is what we have done in the paper for near rings, taking into account the previous results in the field. We study also the pseudohomomorphisms of groups and the applications of near rings to group cohomology.

Identify the elements of a finite dimentional center algebra over field through Wedderburn-Artin's theorem

Yohanes Sukestiyarno, Semarang State University, Indonesia

The paper is concerned with the Wedderburn-Artin's theorem (Passman, 1991). This theorem states that ``if R is a simple Artinian ring, then there exists a unique division ring D and a positive integer number n such that R @ Dn''. The Change problem of R in this theorem into other ring is discussed. The Changing of the simple Artinian R into A a finite dimensional center simple algebra over a field F will be presented so that the isomorphic of A and Dn can still be fulfiled.

The next presentation will be shown that a finite dimentional center algebra D over a field real number \mathbb R and \mathbb R itself, or D and integer C, or D and a quaternion Q over \mathbb R are isomorphic. Based on this result, we have Dn and Mn×n(\mathbb R), Dn and Mn×n(C), or Dn and Mn×n(Q) are isomorphic. So, we regard the elements of Mn×n(\mathbb R), or Mn×n(C), or Mn×n(Q) as the elements of a finite dimentional central simple algebra D over a field \mathbb R.

Key words: Wedderburn-Artin's theorem, finite dimentional center simple algebra

Characterization of Riemannian manifolds by products of some curvature operators

Yulian Tsankov

We characterize the conformal flat or Einstein or foliated manifolds using the following curvature operators :

1. The classical Jacobi operator, 2. The skew-symmetric curvature operator and 3. The Stanilov curvature operator

Small Moufang 2-loops

Petr Vojtechovský, University of Denver, USA

Let M be a finite Moufang loop with normal subloop S such that M/S is cyclic of even order, or dihedral of doubly even order. Then it is possible to modify the multiplication table of M in such a way that the resulting loop (M,*) is again Moufang and, in general, with different center than that of M.

The construction is subtle enough to preserve the associator (as a map, up to equivalence), yet powerful enough to produce all nonassociative Moufang loops of order 16 (5 of them) and of order 32 (71 of them). For n=64, thousands of Moufang loops are obtained. Since the classification of Moufang loops of order 64 is not known, this promises to be a nice tool in describing most (perhaps all) of these 2-loops.

Results on planar near-rings and related near-rings

Gerhard Wendt, Joh. Kepler Univ. Linz

We study planar near-rings and near-rings which have similar properties and give structure results. In particular, we describe planar near-rings (and near-rings with similar properties) as certain centralizer sandwich near-rings and we will consider possible links to the general structure theory of (primitive) near-rings.

When are homogeneous maps linear? A lattice point of view

Marcel Wild, Stellenbosch, South Africa

Call a R-module V Fuchs-Maxson-Pilz (FMP) if for every R-module W every homogeneous map f from V to W is automatically additive. Let L(V) be the modular lattice of all submodules of V. We find necessary respectively sufficient lattice-theoretic properties for being FMP. For instance, each FMP module V has each join irreducible element of L(V) strictly contained in a cyclic submodule. If one restricts attention to b i j e c t i v e homogeneous maps then interesting connections with projective geometry arise.

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