A note on G-type rings

Document Type : Original Paper

Authors

1 Petroleum University of Technology: Ahvaz, Khozestan, 2016-01-21 to present

2 Petroleum University of Technology: Ahvaz,, Ahvaz, Iran

Abstract

In this article, we introduce and examine the concept of $G$-type rings. A ring $R$ is classified as a $G$-type ring if its total quotient ring, denoted by $Q$, is generated by a countable number of elements over $R$ as an $R$-algebra. Formally, $Q = R[S^{-1}]$, where $S$ is a countable set of regular elements in $R$. We establish that $R$ is $G$-type if and only if there exists a countable set of regular elements, denoted as $S$, such that every prime ideal disjoint from $S$ consists solely of zero-divisors.
It is shown that
whenever a ring $T$ is countably generated over a subring $R$, as
an $R$-algebra, and $T$ is strongly algebraic over $R$, then $R$
is $G$-type if and only if $T$ is $G$-type.

Highlights

[1] Artin, E. and Tate, J. T., 1951. A note on finite ring extensions. Journal of the Mathematical Society of Japan, 3, pp. 74-77.
[2] Adams, J. C., 1974. Rings with a finitely generated total quotient ring. Canad. Math. Bull. 17, pp. 1-4.
[3] Dobbs, B. E., Fedder, R. and Fontana, M., 1988. G-domains and Spectoral spaces. J. Pure Appl.
Algebra. 51, pp. 89-110.
[4] Fujita, K., 1975. A note on G(a)-domains and Hilbert rings. Hiroshima Math. J. 5, pp. 61-67.
[5] Goldman, O., 1951. Hilbert Rings and the Hilbert Nullstellensatz. Mathematische Zeitschrift 54,
pp. 136-140.
[6] Kaplansky, I., 1974. Commutative Rings (revised edition). The University of Chicago Press, Chicago.
[7] Karamzadeh, O.A.S. and Moslemi, B., 2012. On G-type domains. Journal of Algebra and its Application. 11, pp. 1-18. DOI: 10.1142/S0219498811005294.
[8] Krull, W., 1951. Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimensionstheorie. Math. Z. 54, pp. 354-387.
[9] Moslemi, B., 2006. On G-domains. Far East J. Math. Sci. (FJMS). 23, pp. 355-359.
[10] Picavet, G., 1975. Autour des id´eaux premiers de Goldman dun anneau commutatif. Ann. Sci. Univ. Clermont Math. 57, pp. 73-90.
[11] Sharp, R.Y. and Vamos, P., 1985. Baire’s category theorem and prime avoidance in complete local rings. Arch. Math. (Basel). 44, pp. 243-248. 

Keywords

Main Subjects

References

[1] Artin, E. and Tate, J. T., 1951. A note on finite ring extensions. Journal of the Mathematical Society of Japan, 3, pp. 74-77.
[2] Adams, J. C., 1974. Rings with a finitely generated total quotient ring. Canad. Math. Bull. 17, pp. 1-4.
[3] Dobbs, B. E., Fedder, R. and Fontana, M., 1988. G-domains and Spectoral spaces. J. Pure Appl.
Algebra. 51, pp. 89-110.
[4] Fujita, K., 1975. A note on G(a)-domains and Hilbert rings. Hiroshima Math. J. 5, pp. 61-67.
[5] Goldman, O., 1951. Hilbert Rings and the Hilbert Nullstellensatz. Mathematische Zeitschrift 54,
pp. 136-140.
[6] Kaplansky, I., 1974. Commutative Rings (revised edition). The University of Chicago Press, Chicago.
[7] Karamzadeh, O.A.S. and Moslemi, B., 2012. On G-type domains. Journal of Algebra and its Application. 11, pp. 1-18. DOI: 10.1142/S0219498811005294.
[8] Krull, W., 1951. Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimensionstheorie. Math. Z. 54, pp. 354-387.
[9] Moslemi, B., 2006. On G-domains. Far East J. Math. Sci. (FJMS). 23, pp. 355-359.
[10] Picavet, G., 1975. Autour des id´eaux premiers de Goldman dun anneau commutatif. Ann. Sci. Univ. Clermont Math. 57, pp. 73-90.
[11] Sharp, R.Y. and Vamos, P., 1985. Baire’s category theorem and prime avoidance in complete local rings. Arch. Math. (Basel). 44, pp. 243-248. 

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History

  • Receive Date: 09 March 2024
  • Revise Date: 14 July 2024
  • Accept Date: 19 February 2025

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