# On common properties of Bezout domains and rings of matrices over them II

- Written by V. Shchedryk
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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 commutative Bezout domain and D∈ M _{n}(R). Denote Ann^{r}(D) by the set of right annihilator of the matrix D:*

*Ann ^{r}(D)={ Q∈ M_{n}(R) | DQ=0 }*

*Denote (A,B) _{l} by and [A,B] the left g.c.d. and right l.c.m. of matrices A, B respectively.*

**Theorem 1.*** Let A,B∈ M _{n}(R). There exists invertible matrix such that *

*where D=(A,B) _{l} and*

*Ann ^{r}(D)⊆ Ann^{r}(V).*

**Theorem 2.*** The matrix is associated by right to matrix *

**Corollary 1.*** The multiple of not zero diagonal elements of right Hermite normal form of matrices A, B is coincide with multiple of not zero diagonal elements of right Hermite normal form of (A,B) _{l} , [A,B]_{r}. *

**Corollary 2.**

*det(AB)=det(A,B) _{l }det[A,B]_{r}.*

* *

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