Which Fields Have No Maximal Subrings?

Abstract

-Fields which have no maximal subrings are completely determined.We observe that the quotient fields of non-field domains have maximal subrings. It is shown that for each non-maximal prime ideal P in a commutative ring R, the ring R P has a maximal subring. It is also observed that if R is a commutative ring with jMax(R)j > 2 d0 or jR=J(R)j > 2 2 d 0 , then R has a maximal subring. It is proved that the well-known and interesting property of the field of the real numbers R (i.e., R has only one nonzero ring endomorphism) is preserved by its maximal subrings. Finally, we characterize submaximal ideals (an ideal I of a ring R is called submaximal if the ring R=I has a maximal subring) in the rings of polynomials in finitely many variables over any ring. Consequently, we give a slight generalization of Hilbert's Nullstellensatz.

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.…”

“…In particular, it is shown that a ring T is submaximal if and only if T is finitely generated as a ring over a proper subring, see [3]. Fields and artinian (non)submaximal rings are completely determined in [6] and [7], respectively. In [8], it is proved that every noetherian ring R with R > 2 ℵ 0 and every infinite direct product of rings are submaximal.…”

“…All subrings, ring extensions, homomorphisms and modules are unital. A proper subring S of a ring R is called a maximal subring if S is maximal with respect to inclusion in the set of all proper subrings of R. A ring with maximal subrings is called a submaximal ring in [4,6,9]. We remind the reader that if S is a maximal subring of a ring R, then the ring extension S ⊆ R is called a minimal ring extension.…”

“…Now, let T = T A N B be the set which consists of 1 and all natural numbers ab such that a b ∈ , and prime divisors of b in B and a has the form Downloaded by [Monash University Library] at 09:01 26 November 2014 THE SPACE OF MAXIMAL SUBRINGS 809 p k i 1 i 1 · · · p k i r i r , where i j ∈ I and k i j ≤ n i j . Then by [6,Remark 1.7], there is a one-one order preserving correspondence between these subsets T of and the subfields of the algebraic closure of p , for each prime number p, taking T into T p = n∈T p n , where p n is the unique subfield of the algebraic closure of p with exactly p n elements. These subsets are called FG-sets in [6].…”

“…For the final aim in this section, i.e., characterizing fields E, for which RgMax E is compact, we need some observations from [6], for the structure of subfields of the algebraic closure of finite fields. Let A = p i i∈I and B = q j j∈J be disjoint subsets of (note, is the set of all prime numbers) and N = n i i∈I ⊆ ∪ 0 .…”