Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
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 aiR, i=1,2,...,n, is uniquely generated of a commutative ring R, if whenever aiR=biR there exist elements uiє R such that ai=biui, i=1,2,...,n, such that ai=biui and u1R+u2R+...+unR=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.
Zabavsky B.V. Diagonal reduction of matrices over rings // Mathematical Studies, Monograph Series, volume XVI, VNTL Publishers, 2012. Lviv. 251 pp.
- Almost zip Bezout domain
- Adequate properties of the elements with almost stable range 1 of a commutative elementary divisor domain
- Bezout rings of stable range 2 and square stable range 1
- A pseudo-irreducible elements of a commutative domain
- Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals