#### Factorization into radical ideals Bezout domains

October 6, 2017 at 15:05 in Lecture Room 379

We characterize the commutative Bezout domains for which has no critical maximal ideals

October 6, 2017 at 15:05 in Lecture Room 379

We characterize the commutative Bezout domains for which has no critical maximal ideals

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

Let A,B∈ Mn(R). There exists invertible matrix such that where D=(A,B)l and Annr(D)⊆ Annr(V).

#### A stable range of class full matrices over elementary divisor ring

Theorem 1. Let R be a commutative Bezout ring a stable range 2. Let A, B∈ F(R2) be such that AR2+BR2=R2, moreover, the matrix B admits a diagonal reduction. Then there exists a full matrix T∈ F(R2) such that A+BT is an invertible matrix. Theorem 2. Let R be a commutative elementary divisor ring. If A, B∈ F(R2) and AR2+BR2=R2, then there exists full matrix T∈ F(R2) such that A+BT is an invertible matrix.

#### Stable rang of commutative domains of elementary divisors

It will be shown that the commutative Bezout domain is an elementary divisor ring if and only if its stable rang equal to half. If commutative Bezout domain is an elementary divisor ring rhen some localization of matrix ring over it is an exchange ring

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

Let R be a commutative Bezout domain in which for any three relatively prime nonzero elements a, b, c there exist such element r, that (a+br,c)=1. And let A1, A2, A3 be relatively prime on left side nonzero matrices from M2(R). Then there exist such matrix T and i,j,k ∈ , that Ai+AjT, Ak are relatively prime on left side

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Department of Algebra and Logic

Faculty of Mechanics and Mathematics

Ivan Franko National University of L'viv

1 Universytetska Str., 79000 Lviv, Ukraine

Tel: (+380 322) 394 172

E-mail: oromaniv at franko.lviv.ua