The Boundary Equivalence for Rings and Matrix Rings over Them

Abstract

We study into the question of whether some rings and their associated matrix rings have equal decidability boundaries in the scheme and scheme-alternative hierarchies. Let $$B_H (A;\sigma )$$ be a decidability boundary for an algebraic system 〈 A; σ 〉 w.r.t. the hierarchy H. For a ring R, denote by $$\underline M _n (R)$$ an algebra with universe $$\bigcup\limits_{1 \leqslant k,l \leqslant n} {R^{k \times l} } $$ . On this algebra, define the operations + and ⋅ in such a way as to extend, if necessary, the initial matrices by suitably many zero rows and columns added to the underside and to the right of each matrix, followed by “ordinary” addition and multiplication of the matrices obtained. The main results are collected in Theorems 1-3. Theorem 1 holds that if R is a division or an integral ring, and R has zero or odd characteristic, then the equalities $$\mathcal{B}_S \left( {R; + , \cdot } \right) = \mathcal{B}_S \left( {R^{n \times n} ; + , \cdot } \right){\text{ and }}\mathcal{B}_S \left( {R; + , \cdot ,1} \right) = \mathcal{B}_S \left( {R^{n \times n} ; + , \cdot ,1} \right)$$ hold for any n⩾1. And if R is an arbitrary associative ring with identity then $$\mathcal{B}_S \left( {R; + , \cdot ,1} \right) = \mathcal{B}_S \left( {R^{n \times n} ;\sigma _0 \cup \left\{ {e_{ij} } \right\}} \right)$$ for any n ⩾ 1 and i,j ∈ { 1,..., n}, where e ij is a matrix identity. Theorem 2 maintains that if R is an associative ring with identity then $$\mathcal{B}_S \left( {\underline M _n \left( R \right)} \right) = \mathcal{B}_S \left( {R; + , \cdot } \right)$$ . Theorem 3 proves that $$\mathcal{B}_{SA} \left( {\underline M _n \left( \mathbb{Z} \right)} \right) = \left\{ {\forall \neg \vee ,\exists \neg \wedge ,\forall \exists ,\exists \forall } \right\}$$ for any n ⩾ 1.