Narrow Results

We determine entirely which Artinian rings have maximal subring. In particular, we show that an Artinian ring without maximal subring is integral over some finite subring and in particular that every Artinian ring which is uncountable or of characteristic zero has a maximal subring. We also determine when a finite direct product of rings has a maximal subring. Finally, we show that if a ring R has an Artinian maximal subring then R itself is Artinian.

It is shown that RgMax(R) is infinite for certain commutative rings, where RgMax(R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then |RgMax(R)| ≥ |Irr(D) ∩ U(R)|, where Irr(D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or |RgMax(R)| ≥ ℵ₀. It is proved that if R is a zero-dimensional (or semilocal) ring with |RgMax(R)| < ℵ₀, then R has nonzero characteristic, say n, and R is integral over ℤ_n. In particular, it is shown that if R is an uncountable artinian ring, then |RgMax(R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2^ℵ₀, then |RgMax(R)| ≥ 2^ℵ₀. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R₂ ⊂ R₁ ⊂ R₀ = R where each R_i is a maximal subring of R_i-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

In this note, we generalize the results of [Submaximal integral domains, Taiwanese J. Math. 17(4) (2013) 1395–1412; Which fields have no maximal subrings? Rend. Sem. Mat. Univ. Padova 126 (2011) 213–228; On the existence of maximal subrings in commutative artinian rings, J. Algebra Appl. 9(5) (2010) 771–778; On maximal subrings of commutative rings, Algebra Colloq. 19(Spec 1) (2012) 1125–1138] for the existence of maximal subrings in a commutative noetherian ring. First, we show that for determining when an infinite noetherian ring R has a maximal subring, it suffices to assume that R is an integral domain with |R/I| < |R| for each nonzero ideal I of R. We determine when the latter integral domains have maximal subrings. In particular, we show that every uncountable noetherian ring has a maximal subring.

A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2^ℵ₀, then |Max(R)| ≤ |RgMax(R)|, where RgMax(R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated.

It is shown that if R is a ring with unit element which is not algebraic over the prime subring of R, then R has a maximal subring. It is shown that whenever R ⊆ T are rings such that there exists a maximal subring V of T, which is integrally closed in T and U(R) ⊈ V, then R has a maximal subring. In particular, it is proved that if R is algebraic over ℤ and there exists a natural number n > 1 with n ∈ U(R), then R has a maximal subring. It is shown that if R is an infinite direct product of certain fields, then the maximal ideals M for which R_M (R/M) has maximal subrings are characterized. It is observed that if R is a ring, then either R has a maximal subring or it must be a Hilbert ring. In particular, every reduced ring R with |R|>2^2ℵ₀ or J(R) ≠ 0 has a maximal subring. Finally, the semi-local rings having maximal subrings are fully characterized.