BZS Near-rings

A right near-ring (N, +,center dot) is called Boolean - zero square or BZS if, for all n is an element of N either n2 = n or n2 = 0. BZS near-rings generalize Boolean, zero square, and Malone trivial near-rings. This paper initiates a study of the structure of BZS nearrings; in particular, it is shown that non-Boolean BZS near-rings with additive groups of prime order must be zero-symmetric Malone trivial near-rings. This leads to results giving both the number of multiplications and the number of isomorphism classes of non-Boolean BZS near-rings with additive groups of prime order. Such near-rings are also shown to have trivial centers provided they have a non-zero multiplication. Examples are given to illustrate and delimit the theory.