# A commutative morphic ring of stable range 2

- Written by Bogdan Zabavsky
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Professor

Department of Algebra and Logic

Faculty of Mechanics and Mathematics

Ivan Franko National University of L'viv

**Abstract:**

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*It is know that a left quasi-morphic ring R is a ring of stable range 1 if and only if dim R=0. *

*(Developing Kaplansky ideas Canfell to introduced the concept of the set of principal ideals a _{i}R, i=1,2,...,n, is uniquely generated of a commutative ring R, if whenever a_{i}R=b_{i}R there exist elements u_{i}є R such that a_{i}=b_{i}u_{i}, i=1,2,...,n, such that a_{i}=b_{i}u_{i} and u_{1}R+u_{2}R+...+u_{n}R=R. The dimension of a commutative ring R, denoted by dim R is the least integer n such that every set of n+1 principal ideals is unique generated.) *

**Theorem.** A commutative morphic ring R is a ring of stable range 2 if and only if dim R=1.

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References

Zabavsky B.V. **Diagonal reduction of matrices over rings** // Mathematical Studies, Monograph Series, volume XVI, VNTL Publishers, 2012. Lviv. 251 pp.

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