Near-rings with right identity

One of the topics algebra is concerned with is computing with objects as if they were numbers. If these objects can be added and subtracted one speaks of a group. If furthermore one can multiply these objects as one is used with the integers one speaks of a ring. If one still can multiply, but can multiply out brackets only from the right hand side and not from the left hand side, one speaks of a near-ring. If there is an object which under multiplication behaves like an identity 1, then one speaks of a near-ring with identity. These near-rings can be fully described as functions mapping from a group into itself. Addition is the usual addition of functions and multiplication is function composition. If one wants to get all near-rings with identity that way, one cannot always take all functions mapping from the group into itself. One has to select functions in a clever way. This is done by the so called centralizer property. Near-rings one does not know so good are near-rings which do not have an identity. These near-rings can be very interesting because they allow unusual methods of computing. Most of these near-rings still have something like a half sided identity. This element behaves like an identity when it is multiplied from the right hand side but not when it is multiplied from the left hand side. Often, this results in interesting methods of computing. In particular, many interesting classes of near-rings are of that type, for example planar near-rings - with a lot of applications inside and outside of algebra - and primitive near-rings, the smallest building stones any near-ring is built of in some sense. Nevertheless, there is no systematic study of near-rings with only a half sided identity up to now. One reason for that is that there has not been an efficient method to describe them so far. With the help of functions mapping from a group only into a subset of this group and together with a new multiplication, the so called sandwich multiplication combined with the centralizer property, one can get an efficient method to describe all near-rings with a half sided identity. This method was developed in the author`s dissertation and was successfully used in some forthcoming papers. However, a deeper research in that direction has not been done so far. This will be done in this project.

Near-rings arise naturally when computing with functions. We compute with functions as if they were numbers, but we cannot follow the same rules as we are used to do when computing with real numbers. The structure of near-rings is considered to be well known especially when they contain an identity element or they satisfy a certain finiteness condition. In this project we were concerned with questions about the structure of near-rings which do not necessarily contain an identity element and do not necessarily satisfy the special finiteness condition. For certain special and important classes of near-rings without identity element we were able to completely determine their structure, thus generalising results which were well known for the case of near-rings containing an identity element. For example, we were able to explicitely give a construction method for primitive near-rings which do not even contain a half sided identity. In doing so, we had to use a special type of multiplication, the so called sandwich multiplication. Primitive near-rings are important in the structure theory of near-rings because each near-ring can be built from primitive near-rings. This put us in a position to further study the structure of near-rings which are closely related to primitive near-rings as well as to study primitive near-rings in their own right. For example, we could obtain many results concerning so called minimal ideals and minimal left ideals in near-rings, thus generalising results which were only known in the case of near-rings with identity or under special (more restrictive) finiteness conditions. Primitive near-rings can also be used to determine the structure of so called automatas, a structure which often arises in theoretical questions in the field of informatics and mathematics. We could, together with a co-author, determine some fundamental results concerning the structure of such automatas using structure theory of (primitive) near-rings. Another class of near-rings which has gained a lot of interest because of its applications is the class of planar near-rings. They can be used for constructing designs for statistical experiments. Planar nearrings do not contain an identity element, unless they are so called near-fields. Also, finite planar near-rings are well understood. Here, together with co-authors, we were able to determine the structure of planar near-rings which are infinite, their additive group is the higher dimensional group of the real numbers and study questions which are linked to topology, another field in mathematics. These considerations also triggered results for classifying special type of functions on vector spaces. Planar near-rings with identity element are near-fields. Near-fields are near-rings where we can divide with each non-zero number and they have a lot of applications in the field of geometry, for example. We were asking if other near-rings exist where the numbers which can be divided form a subnear-field of the near-ring. Surprisingly such near-rings exist, we could determine a lot of their structure and they seem to allow applications similar to that near-fields have. Finally, we were also concerned about near-vector spaces, a generalisation of vector spaces over fields.