Avoidable rings

B. M. Kuznitska; B. V. Zabavsky

doi:10.15330/ms.43.2.153-155

In this article we introduce a new class of rings, so called avoidable rings, which are gene- ralizations of adequate rings and neat rings while in these rings every nonzero prime ideal is contained in a unique maximal ideal.

1. Helmer O. The elementary divisor theorem for certain rings without chain condition// Bull. Amer. Math. Soc. – 1943. – V.49. – P. 225–236.

2. Kaplansky I. Elementary divisors and modules// Trans. Amer. Math. Soc. – 1949. – V.66. – P. 464–491.

3. Gillman L., Henriksen M. Rings of continuous functions in which every finitely generated ideal is principal// Trans. Amer. Math. Soc. – 1956. – V.82. – P. 366–391.

4. Gillman L., Henriksen M. Some remarks about elementary divisor rings// Trans. Amer. Math. Soc. – 1956. – V.82. – P. 362–365.

5. Larsen M.D., Levis W.J., Shores T. Elementary divisor rings, finitely presented modules// Trans. Amer. Math. Soc. – 1974. – V.187. – P. 231–248.

6. Brewer J.W., Conrad P.F., Montgomery P.R. Lattice-ordered groups and a conjecture for adequate domains// Proc. Amer. Math. Soc. – 1974. – V.43, №1. – P. 31–35.

7. McGovern W.Wm. Neat rings// J. Pure Appl. Algebra. – 2006. – V.205. – P. 243–265.

8. Nicholson W.K. Lifting idempotents and exchange rings// Trans. Amer. Math. Soc. – 1977. – V.229. – P. 269–278.

9. Brandal W. Almost maximal integral domains and finitely generated modules// Trans. Amer. Math. Soc. – 1973. – V.183. – P. 203–222.

10. Zabavsky B.V. Diagonal reduction of matrices over rings. – Mat. Studies, Monograph Series, XVI, VNTL, Lviv, 2012. – 251 p.

11. Zabavsky B.V. Diagonal reduction of matrices over finite stable range rings// Mat. Stud. – 2014. – V.41, №1. – P. 101–108.

	Avoidable rings (in Ukrainian)
Author	B. M. Kuznitska, B. V. Zabavsky kuznitska@ukr.net, zabavskii@gmail.com Ivan Franko National University of Lviv
Abstract	In this article we introduce a new class of rings, so called avoidable rings, which are gene- ralizations of adequate rings and neat rings while in these rings every nonzero prime ideal is contained in a unique maximal ideal.
Keywords	avoidable ring; Bezout domain; clean ring; adequate domain
DOI	doi:10.15330/ms.43.2.153-155
Reference	1. Helmer O. The elementary divisor theorem for certain rings without chain condition// Bull. Amer. Math. Soc. – 1943. – V.49. – P. 225–236. 2. Kaplansky I. Elementary divisors and modules// Trans. Amer. Math. Soc. – 1949. – V.66. – P. 464–491. 3. Gillman L., Henriksen M. Rings of continuous functions in which every finitely generated ideal is principal// Trans. Amer. Math. Soc. – 1956. – V.82. – P. 366–391. 4. Gillman L., Henriksen M. Some remarks about elementary divisor rings// Trans. Amer. Math. Soc. – 1956. – V.82. – P. 362–365. 5. Larsen M.D., Levis W.J., Shores T. Elementary divisor rings, finitely presented modules// Trans. Amer. Math. Soc. – 1974. – V.187. – P. 231–248. 6. Brewer J.W., Conrad P.F., Montgomery P.R. Lattice-ordered groups and a conjecture for adequate domains// Proc. Amer. Math. Soc. – 1974. – V.43, №1. – P. 31–35. 7. McGovern W.Wm. Neat rings// J. Pure Appl. Algebra. – 2006. – V.205. – P. 243–265. 8. Nicholson W.K. Lifting idempotents and exchange rings// Trans. Amer. Math. Soc. – 1977. – V.229. – P. 269–278. 9. Brandal W. Almost maximal integral domains and finitely generated modules// Trans. Amer. Math. Soc. – 1973. – V.183. – P. 203–222. 10. Zabavsky B.V. Diagonal reduction of matrices over rings. – Mat. Studies, Monograph Series, XVI, VNTL, Lviv, 2012. – 251 p. 11. Zabavsky B.V. Diagonal reduction of matrices over finite stable range rings// Mat. Stud. – 2014. – V.41, №1. – P. 101–108.
Pages	153-155
Volume	43
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Avoidable rings (in Ukrainian)