The Boundary Equivalence for Rings and Matrix Rings over Them

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 be a decidability boundary for an algebraic system A; w.r.t. the hierarchy H. For a ring R, denote by an algebra with universe . 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 hold for any n1. And if R is an arbitrary associative ring with identity then 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 . Theorem 3 proves that for any n 1.