Partially balanced incomplete block designs from weakly divisible nearrings

In [[6] Riv. Mat. Univ. Parma 11 (2) (1970) 79-96] Ferrero demonstrates a connection between a restricted class of planar nearrings and balanced incomplete block designs. In this paper, bearing in mind the links between planar nearrings and weakly divisible nearrings (wd-nearrings), first we show the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings; consequently, we are able to give some formulas for calculating the design parameters.     

Graph of a Nearring with Respect to an Ideal

We introduce a concept called the graph of a nearring N with respect to an ideal I of N denoted by G _I (N). Then we define a new type of symmetry called ideal symmetry of G _I (N). The ideal symmetry of G _I (N) implies the symmetry determined by the automorphism group of G _I (N). We prove that if I is a 3-prime ideal of a zero-symmetric nearring N then G _I (N) is ideal symmetric. Under certain conditions, we find that if G _I (N) is ideal symmetric then I is 3-prime. Finally, we deduce that if N is an equiprime nearring then the prime graph of N is ideal symmetric.