Diophantine properties of finite commutative rings (Q1407594)

Over some particular rings (for example, over the ring of integers, \(\mathbb{Z }\)) the main logical relations (disjunctions, conjunctions and negations) of polynomial equations admit Diophantine definitions. The main contribution of this paper is to investigate the Diophantine definability of these relations over an arbitrary finite commutative ring with identity. Following this direction there are obtained results and examples, especially in the case of the ring \(\mathbb{Z}/n\mathbb{Z},\) \(n\in \mathbb{N}\). There are also obtained some remarkable new characterizations of finite fields (see, for example, 5.3, 6.1, 6.3 and 7.1).