On when the multiplicative center of a near-ring is a subnear-ring

Summary.

For a near-ring $\langle {N}; {+,\circ}\rangle,$ let $C(N) := \{ c \in N \,|\, x \circ c = c \circ x \text{ for all } x \in N \}$ be its multiplicative center. In contrast to the situation for rings, C(N) need not be a subnear-ring of N. We show that a finite, simple, zero-symmetric near-ring N with $N^{2} \not= \{0\} \text{ and }C(N) \not= \{0\}$ has an identity. If, in addition to these conditions, C(N) is additively closed, then the near-ring $\langle {C(N)}; {+,\circ}\rangle$ is a field.

We describe the multiplicative center of matrix near-rings, and we characterize those near-rings with identity of order p², p a prime, that have additively closed multiplicative center. The latter characterization relies heavily on a construction of the non-rings with identity of order p² from the solutions of the Gołąb-Schinzel functional equation $f(x)f(y) = f(x + y f(x))$ over the field $\mathbb{Z}_p$ .