Finite weakly divisible nearrings

In [5] we dened and studied the algebraic structure called weakly divisible nearring (wd-nearring). In [1, 2] a special class of finite wd-nearrings on Z_{p^n}, p prime, was constructed: on the group (Z_{p^n}; +) of the residue classes (mod p^n) a multiplication "*" can be dened in such a way that (Z_{p^n};+; *) becomes a wd-nearring. Afterwards, in [3, 4] Partially Balanced Incomplete Block Designs (PBIBDs) and codes were obtained starting from the wd-nearrings of [1, 2] and formulae for computing their parameters could be derived just making use of the combinatorial properties of the constructed algebraic structure. In [9] the construction of [1, 2] was generalized to any wd-nearring. Applying Prop. 1 of [9], in Example 2.1 of this paper a wd-nearring N = (Z_7^2 ;+; *) is constructed on the elementary abelian group (Z_7^2 ; +) and a PBIBD is obtained from N. Using the algebraic properties of N = (Z_7^2 ;+; *), all the parameters of the PBIBD are computed. Since it seems reasonable to think the construction and the method to compute all the parameters in [3] could be extended to some additional classes of wd-nearrings, the aim of this paper is to study in more depth the algebraic structure of any finite wd-nearring, especially with regard to determining the size of the elements of signicant structures in N, as partitions, normal chains and products. In the next paragraph, the main definitions and properties of a finite wd-nearring are recalled (Remark 2.1) and the most signicant results presented in this paper are summarized (Remark 2.2).