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

- Written by A. Romaniv
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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 ∈ M _{2}(R). There exists invertible matrices P_{A}, P_{B} and Q_{A}, Q_{B} such that*

*P _{A}AQ_{A}=E, where E=diag(ε_{1},ε_{2}), ε_{1}|ε_{2},*

*P _{B}BQ_{B}=Δ, where Δ=diag(δ_{1},δ_{2}), δ_{1}|δ_{2}.*

*Denote by P_{A} and P_{B} the sets of all matrices P_{A}, P_{B}, 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(ε_{1},0), B∼Δ=diag(δ_{1},0), P_{B}P_{A}^{-1}=||s_{ij}||_{1}^{2}, P_{A}∈ **P**_{A}, P_{B}∈ **P**_{B }. If s_{21}=0, then **P**_{A}∩**P**_{B}≠∅ and (A,B)_{l}=P^{-1}Φ, where Φ=diag((ε_{1},δ_{1}), 0), P∈ **P**_{A}∩**P**_{B}.

**Theorem 2**. * Let A∼E=diag(ε _{1},0), B∼Δ=diag(δ_{1},0), P_{B}P_{A}^{-1}=||s_{ij}||_{1}^{2}, P_{A}∈ P_{A}, P_{B}∈ P_{B }. *

1) If s_{21}=0, then P_{A}∩P_{B}≠∅ and [A,B]_{r}=P^{-1}Ω, where Ω=diag([ε_{1},δ_{1}], 0), P∈ P_{A}∩P_{B};

2) If s_{21}≠0, then [A,B]_{r}=0.