TY - JOUR

T1 - The Boundary Equivalence for Rings and Matrix Rings over Them

AU - Nagrebetskaya, Yu.

PY - 2000

Y1 - 2000

N2 - 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 BH(A;σ) be a decidability boundary for an algebraic system 〈 A; σ 〉 w.r.t. the hierarchy H. For a ring R, denote by M–––n(R) an algebra with universe ⋃1⩽k,l⩽nRk×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 BS(R;+,⋅)=BS(Rn×n;+,⋅) and BS(R;+,⋅,1)=BS(Rn×n;+,⋅,1) hold for any n⩾1. And if R is an arbitrary associative ring with identity then BS(R;+,⋅,1)=BS(Rn×n;σ0∪{eij}) 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 BS(M–––n(R))=BS(R;+,⋅). Theorem 3 proves that BSA(M–––n(Z))={∀¬∨,∃¬∧,∀∃,∃∀} for any n ⩾ 1.

AB - 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 BH(A;σ) be a decidability boundary for an algebraic system 〈 A; σ 〉 w.r.t. the hierarchy H. For a ring R, denote by M–––n(R) an algebra with universe ⋃1⩽k,l⩽nRk×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 BS(R;+,⋅)=BS(Rn×n;+,⋅) and BS(R;+,⋅,1)=BS(Rn×n;+,⋅,1) hold for any n⩾1. And if R is an arbitrary associative ring with identity then BS(R;+,⋅,1)=BS(Rn×n;σ0∪{eij}) 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 BS(M–––n(R))=BS(R;+,⋅). Theorem 3 proves that BSA(M–––n(Z))={∀¬∨,∃¬∧,∀∃,∃∀} for any n ⩾ 1.

UR - http://www.scopus.com/inward/record.url?partnerID=8YFLogxK&scp=27844478689

U2 - 10.1023/A:1010222719323

DO - 10.1023/A:1010222719323

M3 - Article

VL - 39

SP - 396

EP - 406

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 6

ER -