On the standard form for matrices of order two

We establish a criterion for a subring of the field of rational numbers to have a unique standard form (in the sense of Cohn). A similar criterion is obtained for quotient rings of the ring of integers. Definition 1. Let R be an associative ring with unit, C e GL2(R) and C = ^a 0 a₁ 1a₂ 1A a, 1A =L0 eJl-1 0Jl-1 0J...1-1 0J^, where t > 0. Suppose that the following conditions are satisfied: 1) a and e are invertible in A; 2) if 1 < i < t, then a_i is a nonzero non-invertible element of R; 3) if t = 2, then a1 and a2 cannot both be 0. Then the above representation is said to be a standard form for C. Definition 2. 1) A ring R is said to have a unique standard form if no matrix C e GL2R) can be represented by two different standard forms. 2) A ring R is said to be quasi-free if the identity matrix E e GL₂(R) does not possess a nontrivial standard form. Theorem 5. If a ring R is quasi-free, then for every nonzero non-invertible elements b and c of R the element bc - 1 is non-invertible in R. Theorem 5 enables us to prove Proposition 7 and Theorem 8. Proposition 7. Let R = Z/nZ, where n > 1. The following conditions are equivalent: a) R has a unique standard form; b) R is quasi-free; c) n is a prime. Theorem 8. 1) A subring of the field Q is quasi-free if and only if it coincides with Q or with Z. 2) A subring of the field Q has a unique standard form if and only if it coincides with Q. AMS Mathematical Subject Classification: 15A23

Keywords

Authors

References