# Stable range conditions for abelian and duo rings

### Abstract

The article deals with the following question: when does the classical ring of quotients

of a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are there

idempotents in $R$? In the article 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. We find relationships

between the introduced classes of rings and known ones for abelian and duo rings.

We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:

$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1.

The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. We

proved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).

### References

Ara P., Strongly pi-regular rings have stable range one, Proc. Amer. Math. Soc., 124 (1996), 3293–3298.

Auslander M., Goldman O., Maximal orders, Trans. Amer. Math. Soc., 97 (1960), 1–24.

Bass H., K-theory and stable algebra, Publ. Math., 22 (1964), 5–60.

Chen H., Rings with stable range conditions, Comm. Algebra, 26 (1998), 3653–3668.

Cohn P.M., Reversible rings, Bull. London Math. Soc., 31 (1999), №6, 641–648.

Feller E.H., Properties of primary noncommutative rings, Trans. Amer. Math. Soc., 89 (1958), 79–91.

Gillman I., Henriksen M., Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc., 82 (1956), №2, 366–391.

Goodearl K., Menal P., Stable range one for rings with many units, J. Pure Appl. Alg., 54 (1988), 261–287.

Han J., Lee Y., Park S., Semicentral idempotents in a ring, J. Korean Math Soc., 51 (2014), 463–72.

Kaplansky I., Elementary divisor rings and modules, Trans. Amer. Math. Soc., 66 (1949), 464–491.

Kim N.K., Lee Y., Extensions of reversible rings, J. Pure Appl. Alg., 185 (2003), 207–223.

Maury G., Characterisation des ordres maximaux, C. R. Acad. Sc. Paris, 269 (1969), Ser. A, 993–996.

McGovern W.W., Bezout rings with almost stable range 1 are elementary divisor rings, J. Pure and Appl. Algebra, 212 (2007), 340–348.

McGovern W.W., Neat rings, J. Pure Appl. Algebra, 205 (2006), 243–265.

Menal P., Moncasi J., On regular rings with stable range 2, J. Pure and Appl. Algebra, 24 (1982), 25–40.

Lam T.Y., Quasi-duo rings and stable range descent, J. Pure Appl. Alg., 195 (2005), 243–259.

Thierrin G., On duo rings, Canad. Math. Bull., 3 (1960), 167–172.

Tuganbayev A., Automorphism-extendable and endomorphism-extendable modules, Fundam. and Appl. Math., 21 (2016), №4, 175–248. (in Russian)

Zabavsky B., Diagonalizability theorem for matrices over rings with finite stable range, Algebra Discrete Math., 1 (2005), 134–148.

Zabavsky B., Diagonalization of matrices over ring with finite stable range, Visn. of Lviv. Univ., Ser. Mech. Math., 61 (2003), 206–211.

Zabavsky B., Type conditions of stable range for identification of qualitative generalized classes of rings, Algebra Discrete Math., 26 (2018), №3, 140–148.

Zabavsky B.V., Bokhonko V.V. A criterion of elementary divisor domain for distributive domains, Algebra and Discrete Mathematics, 23 (2017), №1, 1–6.

Zabavsky B.V., Gatalevych A.I. Diagonal reduction of matrices over commutative semihereditary Bezout rings, Comm. in Algebra, 47, (2019), 1785–1795.

*Matematychni Studii*,

*57*(1), 92-97. https://doi.org/10.30970/ms.57.1.92-97

Copyright (c) 2022 A. Gatalevych, A. Dmytruk, M. Kuchma

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Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.