Commutative rings with infinitely many maximal subrings (Q2874701)

Let \(R\) be a commutative ring with non-zero identity. A maximal subring of \(R\) is any maximal element of the poset, ordered by inclusion, of all proper unital subrings of \(R\), and RgMax\((R)\) denotes the set, possibly empty, of all maximal subrings of \(R\). The aim of this paper is to find conditions on \(R\) such that RgMax\((R)\) is infinite.NEWLINENEWLINEAmong the results of the paper we mention the following ones: (1) If \(D\) is a subring of \(R\) which is UFD, then \(\,|\,\text{RgMax}(R)\,|\geqslant \text{Irr}(D) \cap U(R)\), where Irr\((D)\) is the set of all non-associate irreducible elements of \(R\) and \(U(R)\) is the set of all invertible elements of \(R\). (2) The ring \(R\) is either Hilbert or \(\,|\,\text{RgMax}(R)\,|\geqslant \aleph _0\). (3) If \(R\) is an uncountable Artinian ring then \(\,|\,\text{RgMax}(R)\,|\geqslant |R|\). (4) If \(R\) is a Noetherian ring with \(\,|R|> 2^{\aleph_0}\) then \(\,\text{RgMax}(R)\,|\geqslant 2^{\aleph_0}\). The authors also determine when a direct product of rings has only finitely many maximal subrings.