On Maximal Subrings of Commutative Rings

Abstract

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| > 2ℵ0, 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.

The Space of Maximal Subrings of a Commutative Ring

“…In fact by the previous comment about the idealization, one can easily see that if K is any field with zero characteristic, then the ring + K is not submaximal, see [9,Example 3.19]. The existence of maximal subrings in commutative rings first studied in [3,4,[6][7][8][9]. In what follows, let us review some needed facts.…”

“…It is interesting to know that every commutative ring R has a minimal ring extension, for if M is a simple R-module, then the idealization R + M is a minimal ring extension of R (note, for any R-module M, every R-subalgebra of R + M has the form R + N , where N is a submodule of M, see [11] or [8,Introduction]). Unlike minimal ring extensions whose existence are guaranteed by the latter result, maximal subrings need not always exist, see [9] for such examples and, in particular, for example of rings of any infinite cardinality, which are not submaximal.…”

Which Fields Have No Maximal Subrings?

“…It is manifest that whenever R contains a field which is either of zero characteristic or uncountable, then every ideal of R is submaximal. We also note that if R is a Noetherian ring with jRj > 2 d 0 , then the nilradical of R is submaximal, by [4,Theorem 2.9]. It is easy to see that a ring R is submaximal if and only if there exist a proper subring S of R and x P R n S such that S[x] R, see [2, Theorem 2.5].…”

“…We remind the reader that whenever R is a maximal subring of a ring T, then either T is integral over R or R is integrally closed in T. Also one can easily see that if R is a maximal subring of T, then the conductor ideal (R : T) is a prime ideal in R, see [8] or [17,Theorem 1 and Theorem 7], see also [4,Remark 3.1]. Moreover, in [8] or [17] it is proved that if R is a maximal subring of a ring T, then T is integral over R if and only if (R : T) P Max(R); it is also shown that if R is integrally closed in T, then (R : T) P Spec(T).…”

“…We denote the set of maximal subrings (maximal ideals) of a ring by RgMax(R) (Max(R)). Maximal subrings are also studied in [17], [5], [6], [14], [13], [16], [2], [3] and [4]. In [2], maximal subrings of commutative rings are investigated and some useful criterions for the existence of maximal subrings are given.…”