Geodesic
Type conditions of stable range for identification of qualitative generalized classes of rings
Algebra and discrete mathematics, Tome 26 (2018) no. 1, pp. 144-152
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This article deals mostly with the following question: when the classical ring of quotients of a commutative ring is a ring of stable range 1? We introduce the concepts of a ring of (von Neumann) regular range 1, a ring of semihereditary range 1, a ring of regular range 1, a semihereditary local ring, a regular local ring. We find relationships between the introduced classes of rings and known ones, in particular, it is established that a commutative indecomposable almost clean ring is a regular local ring. Any commutative ring of idempotent regular range 1 is an almost clean ring. It is shown that any commutative indecomposable almost clean Bezout ring is an Hermite ring, any commutative semihereditary ring is a ring of idempotent regular range 1. The classical ring of quotients of a commutative Bezout ring $Q_{Cl}(R)$ is a (von Neumann) regular local ring if and only if $R$ is a commutative semihereditary local ring.
Keywords: Bezout ring, Hermite ring, elementary divisor ring, semihereditary ring, regular ring, neat ring, clean ring, stable range 1. 
@article{ADM_2018_26_1_a13,
     author = {Bohdan Zabavsky},
     title = {Type conditions of stable range for identification of qualitative generalized classes of rings},
     journal = {Algebra and discrete mathematics},
     pages = {144--152},
     year = {2018},
     volume = {26},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a13/}
}
TY  - JOUR
AU  - Bohdan Zabavsky
TI  - Type conditions of stable range for identification of qualitative generalized classes of rings
JO  - Algebra and discrete mathematics
PY  - 2018
SP  - 144
EP  - 152
VL  - 26
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a13/
LA  - en
ID  - ADM_2018_26_1_a13
ER  - 
%0 Journal Article
%A Bohdan Zabavsky
%T Type conditions of stable range for identification of qualitative generalized classes of rings
%J Algebra and discrete mathematics
%D 2018
%P 144-152
%V 26
%N 1
%U http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a13/
%G en
%F ADM_2018_26_1_a13
Bohdan Zabavsky. Type conditions of stable range for identification of qualitative generalized classes of rings. Algebra and discrete mathematics, Tome 26 (2018) no. 1, pp. 144-152. http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a13/

[1] M. Contessa, “On certain classes of PM-rings”, Comm. Algebra, 12 (1984), 1447–1469 | DOI | MR | Zbl

[2] I. Gillman, M. Henriksen, “Rings of continuous functions in which every finitely generated ideal is principal”, Trans. Amer. Math. Soc., 82:2 (1956), 366–391 | DOI | MR | Zbl

[3] R. Gilmer, J. Huckaba, “$\Delta$-Rings”, J. Algebra, 28 (1974), 414–432 | DOI | MR | Zbl

[4] I. Kaplansky, “Elementary divisors and modules”, Trans. Amer. Math. Soc., 66 (1949), 464–491 | DOI | MR | Zbl

[5] W. McGovern, “Neat rings”, J. Pure and Appl. Algebra, 206:2 (2006), 243–258 | DOI | MR

[6] B. V. Zabavsky, Diagonal reduction of matrices over rings, Mathematical Studies, XVI, VNTL Publishers, Lviv, 2012 | MR | Zbl