Let R be a commutative ring with unity. The notion of maximal non λ-subrings is introduced and studied. A ring R is called a maximal non λ-subring of a ring T if R ⊂ T is not a λ-extension, and for any ring S such that R ⊂ S ⊆ T, S ⊆ T is a λ-extension. We show that a maximal non λ-subring R of a field has at most two maximal ideals, and exactly two if R is integrally closed in the given field. A determination of when the classical D + M construction is a maximal non λ-domain is given. A necessary condition is given for decomposable rings to have a field which is a maximal non λ-subring. If R is a maximal non λ-subring of a field K , where R is integrally closed in K , then K is the quotient field of R and R is a Prüfer domain. The equivalence of a maximal non λ-domain and a maximal non valuation subring of a field is established under some conditions. We also discuss the number of overrings, chains of overrings, and the Krull dimension of maximal non λ-subrings of a field.

中文翻译：

---

最大非 $\lambda$-subrings

令 R 是一个具有统一性的交换环。引入并研究了最大非λ-子环的概念。如果R ⊂ T 不是λ-扩展，并且对于任何满足R ⊂ S ⊆ T 的环S，S ⊆ T 是λ-扩展，则环R 称为环T 的最大非λ-子环。我们证明一个域的最大非 λ 子环 R 至多有两个极大理想，如果 R 在给定域中整体闭合，则恰好有两个。给出了经典 D + M 构造何时是最大非 λ 域的确定。给出了可分解环具有最大非λ-子环场的必要条件。如果 R 是域 K 的最大非 λ 子环，其中 R 在 K 中整体闭合，则 K 是 R 的商域，R 是 Prüfer 域。一个域的一个最大非λ域和一个最大非估价子环的等价是在某些条件下建立的。我们还讨论了一个域的最大非 λ 子环的上环数、上环链和 Krull 维数。