Narrow Results

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.

Let $A$ and $B$ be two rings, $J$ an ideal of $B$ and $f:A\to B$ be a ring homomorphism. The ring $A{\bowtie }^{f}J:=\{(a,f(a)+j)|a\in A,\mathrm{\text{and}}j\in J\}$ is called the amalgamation of$A$with$B$along$J$with respect to$f$. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which $A{\bowtie }^{f}J$, and some related constructions, is either a Hilbert ring, a $G$-domain or a $G$-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the $G$-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

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.