Department of Algebra
Pidstryhach Institute for Applied Problems
of Mechanics and Mathematics
National Academy of Sciences of Ukraine
Let R be an elementary divisor domain, A, B ∈ M2(R). There exists invertible matrices PA, PB and QA, QB such that
PAAQA=E, where E=diag(ε1,ε2), ε1|ε2,
PBBQB=Δ, where Δ=diag(δ1,δ2), δ1|δ2.
Denote by PA and PB the sets of all matrices PA, PB, respectively. Denote by (A,B)l, [A,B]r the greatest common left divisor and the least common right multiple of the matrices A and B.
Theorem1. Let A∼E=diag(ε1,0), B∼Δ=diag(δ1,0), PBPA-1=||sij||12, PA∈ PA, PB∈ PB . If s21=0, then PA∩PB≠∅ and (A,B)l=P-1Φ, where Φ=diag((ε1,δ1), 0), P∈ PA∩PB.
Theorem 2. Let A∼E=diag(ε1,0), B∼Δ=diag(δ1,0), PBPA-1=||sij||12, PA∈ PA, PB∈ PB .
1) If s21=0, then PA∩PB≠∅ and [A,B]r=P-1Ω, where Ω=diag([ε1,δ1], 0), P∈ PA∩PB;
2) If s21≠0, then [A,B]r=0.
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