Model Theory and Near Rings

We recall some basic notions and facts from model theory. Let L be the fi?rst order language of near-rings. Two near-rings R and S are called elementarily equivalent if R and S satisfy the same fi?rst 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 defi?end 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 fi?nitely axiomatisable if can be defi?end by a fi?rst order sentence in L. The fi?nitely axiomatisable classes can be characterized as follows : a class C of near-rings is fi?nitely axiomatisable if it is axiomatisable and the class of near-rings not in C is closed under formation of ultra products.