aequationes mathematicae

, Volume 68, Issue 1–2, pp 46–59 | Cite as

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

  • Erhard Aichinger
  • Mark Farag
Research paper


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 \( \mathbb{Z}_p \) .

Mathematics Subject Classification (2000).

16Y30 39B05. 


Near-ring Center Simple Additively closed Matrix Gołąb-Schinzel Functional equation. 


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Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Institut für Algebra, Stochastik und wissensbasierte mathematische SystemeJohannes Kepler Universität LinzLinzAustria
  2. 2.Department of Mathematics and Computer ScienceWagner CollegeNew YorkUSA

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