Stellens(?)tze For Matrices Over A Commutative Ring

In real algebra geometry, the real Nullstellensatz is a famous theorem, which was first proposed by D.W. Dubois in 1969 and J.J. Risler in 1970 independently. Differing from ordinary algebra geometry, real algebra geometry often involves orderings of ground fields. Because of this reason, the Nichtnegativstellensatz and Positivstellensatz are established in real algebraic geometry as two derivatives of the real Nullstellensatz. Various versions of Stellens(a|¨)tze have been established in the category of real fields.The principal purpose of this paper is to establish correspondingly Stellens(a|¨)tze for matrices over a commutative ring. As a preliminary, we establish several basic facts about eigenvalues of matrices over commutative rings. With the aid of the "abstract" Stellens(a|¨)tze for commutative rings, we establish a Positivstellensatz, a Nullstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. Moreover, as another result in this paper, we further obtain some Stellens(a|¨)tze for matrices over polynomial rings.The Stellens(a|¨)tze in this paper may be viewed as certain generalizations of relevant theorems including the abstract Stellens(a|¨)tze for commutative rings.