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 $\mathcal B_H (A;\sigma)$ be a decidability boundary for an algebraic system $\langle A;\sigma\rangle$ w. r. t. the hierarchy $H$. For a ring $R$ denote by $\underline M_n(R)$ an algebra with universe $\bigcup_{1\leqslant k,l\leqslant n}R^{k\times l}$ On this algebra, define the operations $+$ and $\cdot$ 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 $\mathcal B_S(R;+,\,\cdot\,)=\mathcal B_S(R^{n\times n};+,\,\cdot\,)$ and $\mathcal B_S(R;+,\,\cdot\,,1)=\mathcal B_S (R^{n\times n};+,\,\cdot\,,1)$ hold for any $n>1$. And if $R$ is an arbitrary associative ring with identity then $\mathcal B_S(R;+,\,\cdot\,,1)=\mathcal B_S(R^{n\times n};\sigma_0\cup\{e_{i j}\})$ for any $n\geqslant1$ and $i,j\in\{1,\dots,n\}$, where $e_{ij}$ is a matrix identity.
Theorem 2 maintains that if $R$ is an associative ring with identity then $\mathcal B_S(\underline M_n(R))=\mathcal B_S(R;+,\,\cdot\,)$.
Theorem 3 proves that $\mathcal B_{SA}(\underline M_n(\mathbb Z))=\{\forall\neg\vee,\exists\neg\wedge,\forall\exists,\exists\forall\}$ for any $n\geqslant1$.
