##
**Submaximal integral domains.**
*(English)*
Zbl 1300.13007

The author in [Far East J. Math. Sci. (FJMS) 32, No. 1, 107–118 (2009; Zbl 1164.13004)], and the author collaborating with the reviewer in [Rend. Semin. Mat. Univ. Padova 126, 213–228 (2011; Zbl 1234.13009)], [J. Algebra Appl. 9, No. 5, 771–778 (2010; Zbl 1204.13008)], [Algebra Colloq. 19, Spec. Iss. 1, 1125–1138 (2012; Zbl 1294.13007)] and [Algebra Colloq. 19, Spec. Iss. 1, 1139–1154 (2012; Zbl 1294.13008)], have initiated the study of the existence of maximal subrings in commutative rings \(R\) with \(1\neq 0\). In particular, in [Zbl 1234.13009, Zbl 1204.13008], fields and artinian rings which have maximal subrings are completely characterized, respectively. A commutative ring \(R\) with a maximal subring is called a submaximal ring.

In the article under the review, the author continues his study of latter rings and, in particular, he is interested in domains which are submaximal. In [Zbl 1234.13009], it is observed that the quotient field of any non-field domain is submaximal. In extending this result, he considers a ring \(R\), and a UFD \(D\) as a subring of \(R\) such that \(p^{-1}\in R\) for some irreducible element \(p\in D\), and in Theorem 2.2, shows that \(R\) is submaximal. Consequently, if \(D\) is any non-field UFD, then any ring containing the quotient field of \(D\) is submaximal. In particular, any ring containing the field of rational numbers is submaximal, see also [Zbl 1234.13009, Zbl 1204.13008]. In Corollary 2.19, it is shown that if \(R\) is any domain whose Jacobson radical is nonzero, then any algebra over \(R\) is submaximal which generalizes [Zbl 1294.13008, Corollary 2.24]. Consequently, he observes that for any ring \(R\), either \(R\) is submaximal or it is a Hilbert ring with \(\mathrm{Spec}(R)\leq 2^{2^{\aleph _0}}\). A ring \(R\) is called a residue finite ring if \(\frac{R}{I}\) is a finite ring for every nonzero ideal \(I\) of \(R\). In theorem 2.29, Azarang characterizes residue finite rings which are non-submaximal.

In particular, it is shown that these rings are countable principal ideal rings which are algebraic over their prime subrings.

Moreover, it is observed that if \(R\) is a residue finite ring, which is non-submaximal, then either \(\dim(R)=1\), in which case, \(R\) is a domain generated by its units, or \(\dim(R)=0\), in which case, \(R\) is artinian with nonzero characteristic, whose set of ideals is finite. As a consequence of Theorem 2.31, it’s observed that if \(S\) is a multiplicatively closed subset of \(R\) such that \(R_S\) is submaximal and \(\mathrm{Max}(R_S)=\mathrm{Spec}(R_S)\) is a finite set, then \(R\) is submaximal too. Let us recall that an integral domain \(R\) is called atomic if each nonzero element of \(R\) is finite product of irreducible elements (not necessarily unique) of \(R\) (e.g., noetherian domains). The following three interesting results, of which, the third one is the main result of this article show that certain domains are submaximal.

Theorem A (Theorem 3.5). Let \(R\) be an uncountable atomic domain with zero characteristic such that every natural number has at most countably many divisors in \(R\). Then \(R\) is submaximal.

Theorem B (Proposition 3.10). Let \(R\) be an uncountable Dedekind domain whose set of maximal ideals is countable. Then \(R\) is submaximal.

The next theorem generalizes the fact that every uncountable field is submaximal, see [Zbl 1204.13008, Corollary 1.3].

Theorem C (Theorem 3.1). Every uncountable UFD is submaximal.

Reviewer’s comment: Reading the following forthcoming articles together with the pertinent ones in the list of references of the article under the review, provide a good and useful background for those young students who are looking for some reasonable and rather new topics in commutative rings to do research.

A. Azarang, “The space of maximal subrings of a commutative rings”, to appear in Commun. Algebra, doi:10.1080/00927872.2013.849264.

A. Azarang and G. Oman, J. Algebra Appl. 13, No. 7, Article ID 1450037, 29 p. (2014; Zbl 1308.13012).

A. Azarang, “On the existence of maximal subrings in commutative noetherian rings”, to appear in J. Algebra Appl., doi:10.1142/S021949881450073X.

In the article under the review, the author continues his study of latter rings and, in particular, he is interested in domains which are submaximal. In [Zbl 1234.13009], it is observed that the quotient field of any non-field domain is submaximal. In extending this result, he considers a ring \(R\), and a UFD \(D\) as a subring of \(R\) such that \(p^{-1}\in R\) for some irreducible element \(p\in D\), and in Theorem 2.2, shows that \(R\) is submaximal. Consequently, if \(D\) is any non-field UFD, then any ring containing the quotient field of \(D\) is submaximal. In particular, any ring containing the field of rational numbers is submaximal, see also [Zbl 1234.13009, Zbl 1204.13008]. In Corollary 2.19, it is shown that if \(R\) is any domain whose Jacobson radical is nonzero, then any algebra over \(R\) is submaximal which generalizes [Zbl 1294.13008, Corollary 2.24]. Consequently, he observes that for any ring \(R\), either \(R\) is submaximal or it is a Hilbert ring with \(\mathrm{Spec}(R)\leq 2^{2^{\aleph _0}}\). A ring \(R\) is called a residue finite ring if \(\frac{R}{I}\) is a finite ring for every nonzero ideal \(I\) of \(R\). In theorem 2.29, Azarang characterizes residue finite rings which are non-submaximal.

In particular, it is shown that these rings are countable principal ideal rings which are algebraic over their prime subrings.

Moreover, it is observed that if \(R\) is a residue finite ring, which is non-submaximal, then either \(\dim(R)=1\), in which case, \(R\) is a domain generated by its units, or \(\dim(R)=0\), in which case, \(R\) is artinian with nonzero characteristic, whose set of ideals is finite. As a consequence of Theorem 2.31, it’s observed that if \(S\) is a multiplicatively closed subset of \(R\) such that \(R_S\) is submaximal and \(\mathrm{Max}(R_S)=\mathrm{Spec}(R_S)\) is a finite set, then \(R\) is submaximal too. Let us recall that an integral domain \(R\) is called atomic if each nonzero element of \(R\) is finite product of irreducible elements (not necessarily unique) of \(R\) (e.g., noetherian domains). The following three interesting results, of which, the third one is the main result of this article show that certain domains are submaximal.

Theorem A (Theorem 3.5). Let \(R\) be an uncountable atomic domain with zero characteristic such that every natural number has at most countably many divisors in \(R\). Then \(R\) is submaximal.

Theorem B (Proposition 3.10). Let \(R\) be an uncountable Dedekind domain whose set of maximal ideals is countable. Then \(R\) is submaximal.

The next theorem generalizes the fact that every uncountable field is submaximal, see [Zbl 1204.13008, Corollary 1.3].

Theorem C (Theorem 3.1). Every uncountable UFD is submaximal.

Reviewer’s comment: Reading the following forthcoming articles together with the pertinent ones in the list of references of the article under the review, provide a good and useful background for those young students who are looking for some reasonable and rather new topics in commutative rings to do research.

A. Azarang, “The space of maximal subrings of a commutative rings”, to appear in Commun. Algebra, doi:10.1080/00927872.2013.849264.

A. Azarang and G. Oman, J. Algebra Appl. 13, No. 7, Article ID 1450037, 29 p. (2014; Zbl 1308.13012).

A. Azarang, “On the existence of maximal subrings in commutative noetherian rings”, to appear in J. Algebra Appl., doi:10.1142/S021949881450073X.

Reviewer: O. A. S. Karamzadeh (Ahvaz)