"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

# Common divisors and common multiples of singular matrices over elementary divisor domains

Andrij Romaniv

Department of Algebra
Pidstryhach Institute for Applied Problems
of Mechanics and Mathematics
National Academy of Sciences of Ukraine

Abstract:

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(ε12),  ε12,

PBBQB=Δ,  where Δ=diag(δ12),  δ12.

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,  PAPA, PBPB .  If  s21=0, then  PAPB≠∅  and  (A,B)l=P-1Φ,  where  Φ=diag((ε11), 0),  P∈ PAPB.

Theorem 2. Let A∼E=diag(ε1,0),  B∼Δ=diag(δ1,0),  PBPA-1=||sij||12,  PAPA, PBPB .
1) If  s21=0, then  PAPB≠∅  and  [A,B]r=P-1Ω,  where  Ω=diag([ε11], 0),  P∈ PAPB;
2) If  s21≠0, then  [A,B]r=0.

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O. Romaniv, 27 Apr, 2018

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