Model Theory and Near Rings

We recall some basic notions and facts from model theory. Let L be the first order language of near-rings. Two near-rings R and S are called elementarily equivalent if R and S satisfy the same first order sentences in L: A class C of near-rings is called elementarily closed if R is elementary equivalent to S; S belongs to C then R belongs to C. A class C of near-rings is called axiomatisable if C can be defiend by a family of first order sentences in L. Also a class C is axiomatisable if C is elementarily closed and closed under the formation of ultra products. Further, a class C of near-rings is called finitely axiomatisable if can be defiend by a first order sentence in L. The finitely axiomatisable classes can be characterized as follows : a class C of near-rings is finitely axiomatisable if it is axiomatisable and the class of near-rings not in C is closed under formation of ultra products.