Submaximal Integral Domains

Abstract

It is proved that if D is a U F D and R is a D-algebra, such that U (R) ∩ D = U (D), then R has a maximal subring. In particular, if R is a ring which either contains a unit x which is not algebraic over the prime subring of R, or R has zero characteristic and there exists a natural number n > 1 such that 1 n ∈ R, then R has a maximal subring. It is shown that if R is a reduced ring with |R| > 2 2 ℵ 0 or J(R) = 0, then any R-algebra has a maximal subring. Residually finite rings without maximal subrings are fully characterized. It is observed that every uncountable U F D has a maximal subring. The existence of maximal subrings in a noetherian integral domain R, in relation to either the cardinality of the set of divisors of some of its elements or the height of its maximal ideals, is also investigated.2000 Mathematics Subject Classification. 13B02, 13G05, 13E05.

The Space of Maximal Subrings of a Commutative Ring

“…Similarly, if R is a ring and x is a unit in R which is not algebraic over the prime subring of R, then R has a maximal subring S which does not contain x −1 . Consequently, if R is a nonsubmaximal ring, then U R and therefore J R are algebraic over the prime subring of R. Using the previous result in [4], it is proved that every uncountable UFD is submaximal.…”

“…Consequently, every reduced ring R with R > 2 2 ℵ 0 or J R = 0 is submaximal. Finally, in [4], the following useful result is proved. The previous theorem has several interesting consequences.…”

“…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.…”

“…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.…”

Commutative rings with infinitely many maximal subrings

“…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. Not every ring possesses maximal subrings (for example the algebraic closure of a finite field has no maximal subrings, see [14,Corollary 2.7] or [7,Remark 1.13]; also see [6,Example 2.6] and [9,Example3.19] for more examples of rings which have no maximal subrings). A ring which possesses a maximal subring is said to be submaximal, see [3], [7] and [9]. If S is a maximal subring of a ring R, then the extension S ⊆ R is called a minimal ring extension (see [21]) or an adjacent extension too (see [16]).…”