1 Introduction

All rings and algebras considered below are supposed to be integral domains; all inclusions of rings, ring homomorphisms and algebra homomorphisms are unital. For any ring extension \(R\subset S\), we let [R, S] denote the set of all intermediate rings (i.e., rings T such that \(R\subseteq T \subseteq S\)). Any ring T in [R, S] is called an S-overring of R. Such a ring is called a proper S-overring of R if, in addition, \(T \ne R\). In the particular case when S is the quotient field of R, then each S-overring of R is called an overring of R. Such a ring T is termed a proper overring of R if \(T \ne R\). A ring R is called a maximal non-integrally closed subring of S if R is not integrally closed in S, whereas each proper S-overring of R is integrally closed in S. Jaballah [20] has studied maximal non-integrally closed subrings of a field. The main goal of this paper is to pursue and deepen the study of Jaballah and to characterize maximal non-integrally closed subrings in case the top ring is not necessarily a field. Section 2 is devoted to recall some background concerning normal pairs of rings and minimal ring extension that we will use extensively in the proofs of our results. In Sect. 3, we establish several equivalent statements for maximal non-integrally closed subrings of an integral domain S (see Theorems 3.3 and 3.4). As a consequence, Corollary 3.6 recovers [20, Theorem 5]. The case where \(R\subset S\) is an integral extension is treated in Corollary 3.7. The paper is concluded with several examples to illustrate our theory.

Let \(R\subset S\) be a ring extension. Throughout this paper, qf(R) denotes the quotient field of the integral domain R and Spec(R) (resp., Max(R)) denotes as usual the set of prime (resp., maximal) ideals of R. If \(R\subseteq S\) is a ring extension and P is a prime ideal of R, then \({\overline{R}}_S\) denotes the integral closure of R in S and \(S_P\) denotes the localization of S with respect to the multiplicative closed subset \(R \setminus P\). We use “\(\subseteq \)” for inclusion and “\(\subset \)” for strict inclusion. Any undefined notation or terminology is standard, as in [15, 21].

2 Preliminaries

In this section, we collect some definitions and results concerning normal pairs and minimal extensions of rings. Let \(R\subset S\) be a ring extension. According to Davis [7], the pair (R, S) is called a normal pair if each element in [R, S] is integrally closed in S. The set of intermediate rings in normal pairs and the set of overrings of a Prüfer domain have quite similar properties. For instance, if R is a local domain, then (R, S) is a normal pair if and only if there is a divided prime ideal P of R (that is, \(PR_P = P\)) such that \(S=R_P\) and R/P is a valuation domain (cf. [7, Theorem 1]). Ayache and Jaballah [2] have continued the study of normal pairs of integral domains, so many characterizations of such pairs were established. They have also introduced residually algebraic pairs, a concept which is closely related to normal pairs. Recall that a pair of rings (R, S), where \(R\subset S\) is a ring extension, is called a residually algebraic pair, if for any intermediate ring T between R and S and any prime ideal Q of T, the ring extension \(R/P\subseteq T/Q\) is algebraic, where \(P= Q\cap R\) (see [2, Definition 2.1]). It is worthwhile noticing that (R, S) is a residually algebraic pair if and only if \(({\overline{R}}_S, S)\) is a normal pair (cf. [2, Theorem 2.10]).

Recall that a ring extension \(R\subset S\) is said to be minimal if R is a proper subring of S and \([R, S]=\{R, S\}\). As \({\overline{R}}_S\) is an intermediate ring between R and S, then either \({\overline{R}}_S=R\), in which case R is integrally closed in S; or \({\overline{R}}_S=S\), in which case \(R\subset S\) is an integral extension. In what follows, we will recall some results about minimal ring extensions. According to [13, Théorème 2.2(i) and Lemme 1.3], if \(R\subset S\) is a minimal extension and R is not a field, then there exists a unique maximal ideal M of R called the crucial maximal ideal of \(R\subset S\) such that the canonical injective ring homomorphism \(R_M\rightarrow S_M\) can be viewed as a minimal ring extension, while the canonical ring homomorphism \(R_P\rightarrow S_P\) is an isomorphism for all prime ideals P of R, except M (cf. [8, 13] and [22, p. 37]). If in addition, \(R\subset S\) is an integral extension, then M is precisely the conductor \((R:S):=\{x\in R \mid xS\subseteq R\}\) (see [13, Théorème 2.2(ii)]). Recall from [10, Theorems 2.2 and 2.3] that if \(R\subset S\) is an extension and \(M=(R:S)\), then \(R\subset S\) is a minimal integral extension if and only if \(M\in Max(R)\) and one of the following three conditions holds:

  1. (a)

    inert case: \(M \in Max(S)\) and \(R/M \rightarrow S/M\) is a minimal field extension;

  2. (b)

    decomposed case: There exist \(M_1, M_2 \in Max(S)\) such that \(M = M_1 \cap M_2\) and the natural maps \(R/M \rightarrow S/M_1\) and \(R/M \rightarrow S/M_2\) are both isomorphisms; or, equivalently, there exists \(q \in S \setminus R\) such that \(S = R[q], q^2 - q \in M\), and \(Mq \subseteq M\).

  3. (c)

    ramified case: There exists \(M' \in Max(S)\) such that \(M'^2 \subseteq M \subset M'\), \([S/M : R/M] = 2\), and the natural map \(R/M \rightarrow S/M'\) is an isomorphism; or, equivalently, there exists \(q \in S \setminus R\) such that \(S = R[q], q^2 \in M\), and \(Mq \subseteq M\).

For more details about minimal extensions, see [8, 13, 22]. Recall also that if \(R\subset S\) is a ring extension, a proper S-overring A of R is called the minimal overring of R in S, if \(A\subseteq T\) for any proper S-overring T of R (see [3, 17]).

3 Main results

We start with the following definition.

Definition 3.1

Let \(R\subset S\) be an extension of integral domains. We say that R is a maximal non-integrally closed subring of S if R is not integrally closed in S, whereas T is integrally closed in S for any proper S-overring T of R.

Remark 3.2

If R is a maximal non-integrally closed subring of S, then \(({\overline{R}}_S, S)\) is a normal pair (or equivalently, (R, S) is a residually algebraic pair). Thus, \(R\subset S\) is an algebraic extension.

The following theorem identifies maximal non-integrally closed subrings in case the bottom ring is a field.

Theorem 3.3

Let \(R\subset S\) be an extension of integral domains. If R is a field, then the following statements are equivalent:

  1. (1)

    R is a maximal non-integrally closed subring of S.

  2. (2)

    \(R \subset S\) is a minimal field extension.

Proof

Only the implication (1)\(\Rightarrow \)(2) needs a proof. To this end, suppose that T is a ring such that \(R\subset T \subseteq S\). According to Remark 3.2, the ring extension \(R\subset S\) is algebraic. Hence, S is a field algebraic over R. Therefore, \(T\subseteq S\) is an algebraic field extension. As, by assumption, T is integrally closed in S, then a fortiori \(T=S\). Hence, \(R \subset S\) is a minimal field extension. This completes the proof. \(\square \)

Gilmer and Heinzer [17] asked which domains admit a unique minimal overring. Ben Nasr and Zeidi have answered this key question but with the additional assumption that the ring extension \(R\subset S\) satisfies FCP (see [5, Corollaries 2.4 and 2.9]). Recall that a ring extension \(R\subset S\) is said to satisfy FCP if each chain of distinct intermediate rings of this extension is finite. It is worthwhile noting that if \(R\subset S\) satisfies FCP, then (R, S) is a residually algebraic pair (cf. [19, Theorem 3.9]). But, the converse is false. Indeed, one can consider a non semilocal Prüfer domain R with quotient field K. Then (R, K) is a normal pair, but \(R\subset K\) does not satisfy FCP (see [16, Theorem 1.5]).

Now, we present our main result which characterizes maximal non-integrally closed subrings in case the bottom ring is not a field and answers the above question raised by Gilmer and Heinzer in case \(({\overline{R}}_S, S)\) is a normal pair.

Theorem 3.4

Let \(R\subset S\) be an extension of integral domains such that R is not a field. Then the following statements are equivalent:

  1. (1)

    R is a maximal non-integrally closed subring of S.

  2. (2)

    \({\overline{R}}_S\) is the unique minimal overring of R in S and \(({\overline{R}}_S, S)\) is a normal pair.

  3. (3)

    \(R\subset {\overline{R}}_S\) is a minimal extension, \(({\overline{R}}_S, S)\) is a normal pair and \(R=R_M\cap S\), where \(M=(R:{\overline{R}}_S)\).

Proof

(1)\(\Rightarrow \)(2) Let T be a proper S-overring of R, then T is integrally closed in S. Thus, \({\overline{R}}_S\subseteq {\overline{T}}_S=T\). This shows that \({\overline{R}}_S\) is the unique minimal overring of R in S. The fact that \(({\overline{R}}_S, S)\) is a normal pair follows readily from Remark 3.2.

(2)\(\Rightarrow \)(3):

We only need to prove that \(R=R_M\cap S\). If \(R_M\cap S \ne R\), then \({\overline{R}}_S\subseteq R_M\cap S\) since \({\overline{R}}_S\) is the minimal overring of R in S. Thus \({\overline{R}}_S\subseteq R_M\) and so \({{\overline{R}}_S}_M= R_M\), a contradiction.

(3)\(\Rightarrow \)(1):

By assumption, R is not integrally closed in S. We need to prove that if T is a ring in [R, S] such that \(T\ne R\), then T is integrally closed in S. To this end, notice that the ring \(T\cap {\overline{R}}_S\) lies between R and \({\overline{R}}_S\) and as \(R\subset {\overline{R}}_S\) is a minimal extension, \(T\cap {\overline{R}}_S= R\) or \({\overline{R}}_S\). If \(T\cap {\overline{R}}_S={\overline{R}}_S\), then \({\overline{R}}_S\subseteq T\). Hence T is integrally closed in S since \(({\overline{R}}_S, S)\) is a normal pair. Now, we will show that the case when \(T\cap {\overline{R}}_S = R\) is impossible. Since \(T\cap {\overline{R}}_S = R\), then R is integrally closed in T and as (R, T) is a residually algebraic pair (because (R, S) is a residually algebraic pair), it follows that (R, T) is a normal pair. Note that in general, if A is integrally closed in B and N is a multiplicatively closed subset of A, then \(A_N\) is integrally closed in \(B_N\) (cf. [1, Proposition 5.12]). It is not difficult to check that any intermediate ring between \(A_N\) and \(B_N\) has the form \(H_N\) for some ring H intermediate between A and B. But, for the convenience of the reader, we will provide some details. Indeed, let \(E\in [A_N, B_N]\) and set \(H=E\cap B\). If \(\alpha \in E\), then \(\alpha =\frac{b}{n}\) for some \(b\in B\) and \(n\in N\). Clearly, \(b=n\alpha \in NE\subseteq AE\subseteq E\). Thus, \(b\in E\cap B\) and so \(\alpha \in H_N\). Conversely, let \(\zeta \in H_N\) and write \(\zeta =\frac{h}{s}\), where \(h\in H=E\cap B\) and \(s\in N\). Notice that \(\frac{1}{s}\in A_N\subseteq E\). Hence, \(\zeta =\frac{1}{s} h\in E\), as desired. It follows from the above comments that if (A, B) is a normal pair, then so is the pair \((A_N, B_N)\).

Now, as (R, T) is a normal pair then \((R_M, T_M)\) is also a normal pair. We claim that \(R_M=T_M\). Indeed, assume by way of contradiction that \(R_M\ne T_M\). Since \((R_M, T_M)\) is a normal pair, then there exists a prime ideal Q of \(R_M\) such that \(T_M=(R_M)_Q\) (cf. [7, Theorem 1]). As \(R_M\subset {{\overline{R}}_S}_M\) is a minimal extension, with crucial maximal ideal \(MR_M\), and Q does not contain \(MR_M\), then [6, Proposition 4] ensures that \((R_M)_Q= ({{\overline{R}}_S}_M)_Q\). This implies that \(R_M\subset {{\overline{R}}_S}_M \subseteq T_M\), a contradiction since \(R_M\) is integrally closed in \(T_M\). Therefore, \(R_M=T_M\) as claimed. It follows that \(T\subseteq T_M\cap S=R_M\cap S=R\), a contradiction completing the proof. \(\square \)

Remark 3.5

It follows from the previous theorem that if R is a maximal non-integrally closed subring of S and R is not a field, then qf(R) \(=\) qf(S). Indeed, Theorem 3.4 ensures that \(R\subset ~{\overline{R}}_S\) is a minimal (integral) extension and \(({\overline{R}}_S, S)\) is a normal pair. The rings R and \({\overline{R}}_S\) share the ideal \(M=(R:{\overline{R}}_S)\). Moreover, M is a nonzero ideal. To see this, note that as \(R\subset {\overline{R}}_S\) is a minimal extension, then \(M=(R:{\overline{R}}_S)\) is the crucial maximal ideal of this ring extension and M is a maximal ideal of R. Hence, \(M\ne (0)\) since R is not a field. Now, according to [6, Proposition 0], we get easily qf(R) \(=\) qf(\({\overline{R}}_S\)). On the other hand, since \(({\overline{R}}_S, S)\) is a normal pair, then clearly qf(\({\overline{R}}_S\)) \(=\) qf(S). It follows that qf(R) \(=\) qf(S).

When S is a field, the condition “\(R=R_M\cap S\)” is equivalent to the fact that R is local with maximal ideal M. Thus the following result recovers [20, Theorem 5].

Corollary 3.6

Let \(R\subset S\) be an extension of integral domains such that S is a field and R is not a field. Then the following statements are equivalent:

  1. (1)

    R is a maximal non-integrally closed subring of S.

  2. (2)

    \({\overline{R}}_S\) is the unique minimal overring of R in S and \({\overline{R}}_S\) is a Prüfer domain with quotient field S.

  3. (3)

    \(R\subset {\overline{R}}_S\) is a minimal extension, \({\overline{R}}_S\) is a Prüfer domain with quotient field S and R is local.

The next corollary characterizes maximal non-integrally closed subrings in case the ring extension \(R\subset S\) is integral.

Corollary 3.7

Let \(R\subset S\) be an integral extension such that R is not a field. Then the following statements are equivalent:

  1. (1)

    R is a maximal non-integrally closed subring of S.

  2. (2)

    \(R\subset S\) is a minimal extension.

Next we provide some examples illustrating our theory. But first, we recall the concept of pullback constructions. Let I be a nonzero ideal of an integral domain T, \(\varphi : T\longrightarrow E:=T/I\) the natural projection, and D an integral domain contained in E. Then let \(R=\varphi ^{-1}(D)\) be the integral domain arising from the following pullback of canonical homomorphisms:

$$\begin{aligned} \begin{array}{ccc} R &{} \rightarrow &{} D \\ \downarrow &{} &{} \downarrow \\ T &{} \rightarrow &{} T/I \end{array} \end{aligned}$$

We shall assume that R is properly contained in T. For more details about pullbacks, one can consult [6]. The following three examples are an application of Corollaries  3.6 and  3.7.

Example 3.8

This example provides a ring extension \(R\subset S\) of integral domains such that R is a maximal non-integrally closed subring of S and \(R\subset {\overline{R}}_S\) is an inert extension, but R is not a maximal non-integrally closed subring of its quotient field.

Let S be a domain with exactly two maximal ideals M and N such that \(S/M\cong S/N \cong L\). Let K be a subfield of L such that \(K\subset L\) is a minimal field extension. Let \(R= \varphi ^{-1}(K)\), where \(\varphi : S\longrightarrow S/M\) is the natural projection. Then R is a semilocal domain with exactly two maximal ideals, namely M and \(N\cap R\). As \(R/M\subset S/M\) is a minimal field extension, then \(R\subset S\) is a minimal integral extension. Thus \(R\subset {\overline{R}}_S=S\) is an inert extension. It follows from Corollary 3.7 that R is a maximal non-integrally closed subring of S. Corollary 3.6 ensures that R is not a maximal non-integrally closed subring of its quotient field, since R is not local.

Example 3.9

This example provides a ring extension \(R\subset S\) such that R is a maximal non-integrally closed subring of S, \(R\subset {\overline{R}}_S\) is a decomposed extension and R is not a maximal non-integrally closed subring of its quotient field.

Let S be a domain with exactly three maximal ideals M, N and \(M'\) such that \(S/M\cong S/N \cong S/M'\cong L\). Let \(R= \varphi ^{-1}(L)\), where \(\varphi : S\longrightarrow S/M\cap N\) is the natural projection. Then R is a semilocal domain with exactly two maximal ideals, namely \(M\cap N\) and \(M'\cap R\). As \(R/M\cap N=L\subset S/M\cap N \cong L\times L\) is a minimal integral extension, then \(R\subset S\) is also a minimal integral extension. Hence, \(R\subset {\overline{R}}_S\) is a decomposed extension. It follows from Corollary 3.7 that R is a maximal non-integrally closed subring of S. As R is not local, then R cannot be a maximal non-integrally closed subring of its quotient field by virtue of Corollary 3.6.

Example 3.10

This example provides a ring extension \(R\subset S\) such that R is a maximal non-integrally closed subring of S, \(R\subset {\overline{R}}_S\) is a ramified extension and R is not a maximal non-integrally closed subring of its quotient field.

Let S be a domain with exactly two maximal ideals M and N such that \(S/M\cong S/N \cong L\). Assume that one can write \(S=L+M\). Let \(R=L+M^2\). Then R is a semilocal domain with exactly two maximal ideals, namely \(M^2\) and \(N\cap R\). As \(R/M^2\subset S/M^2\) is a minimal integral extension, then so is \(R\subset S\). Thus \(R\subset {\overline{R}}_S\) is a ramified extension. The ring R is a maximal non-integrally closed subring of S according to Corollary 3.7. On the other hand, Corollary 3.6 implies that R is not a maximal non-integrally closed subring of its quotient field, since R is not local.

Remark 3.11

If we assume in Example 3.8 that S is a Prüfer domain and we set \(T=S_N\), then clearly \({\overline{R}}_T=S\) and (S, T) is a normal pair. On the other hand, as N does not contain M, then [6, Proposition 4] guarantees that \(S_N=R_{N\cap R}\). Hence, \(R_M\cap T=R_M\cap R_{N\cap R}=R\). Now, Theorem 3.4 applies easily to conclude that R is a maximal non-integrally closed subring of T. We can argue similarly for Examples 3.9 and 3.10.

We close the paper with the following two open questions. We use the symbol [t]] to stand for “[t] or [[t]]”.

Question 1

Let \(S = K[y_1]]\ldots [y_t]]\) be a K-algebra (not necessarily finitely generated over the field K) having Krull dimension \(n \ge 1\). Let I be a nonzero proper ideal of S (not necessarily maximal in S) and D be a proper subring of K. Provide necessary and sufficient conditions in order that \(R = D + I\) is a maximal non-integrally closed subring of S.

Question 2

Characterize when R is a maximal non-integrally closed subring of S in case \(R\subset S\) is an extension of commutative rings with zero divisors or more generally in case \(R\subset S\) is an extension of associative rings.

The first question is motivated by two papers. The first one is [23], in which Visweswaran characterizes when \(R = D + I\) is a maximal non-Noetherian subring of the finitely generated algebra \(S=K[y_1,\ldots , y_t]\) and the second one is [14], where Gasmi and Jarboui have provided necessary and sufficient conditions under which R is maximal non-Jaffard subring of the K-algebra \(S = K[y_1]]\ldots [y_t]]\). The second question is motivated by the increasing interest in normal pairs of commutative rings with zero divisors on one hand (cf. [4, 11, 12, 18]) and non commutative rings on the other hand (cf. [9]).