By Lee J. (ed.)
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Additional resources for Commutative rings. New research
14 (a), (b)]). Let I be a index set such that |I| ≥ 2 and Ti a nonzero ring for each i ∈ I. Consider the ring direct product T := ∏i∈I Ti and a ring extension K ⊆ T , where K is a field. If K ⊆ T is a λ-extension, then |I| = 2. Indeed, K ⊆ T is a λ-extension if and only if T is isomorphic as a K-algebra to K × L for some field L such that K ⊆ L is a λ-extension. 10, one may ask what can be said about arbitrary λ-extensions of a field. 11. Recall that dim(A) denotes the Krull dimension of a ring A and dimk (E) denotes the vector space dimension of a vector space E over a field k.
1. 2 (c) generalizes a result of G. Picavet and M. PicavetL’Hermitte  on the minimal overrings of a local Noetherian ring. 30 David E. 2. 2]) Let R be a quasilocal ring and Q a given total ring of quotients of R. If T is a minimal ring extension of R such that R is integrally closed in T , then there exists a unique ring A such that R ⊂ A ⊆ Q and T ∼ = A as R-algebras. Thus, there do not exist distinct R-subalgebras A1 , A2 of Q such that R ⊂ Ai is a minimal ring extension and R is integrally closed in Ai for i = 1, 2.
Rn = T and Ri ⊂ Ri+1 is a minimal ring extension for each i = 0, . . , n − 1. The pedigree of the FIP concept is stalwart and algebraic, for the classical Primitive Element Theorem can be reformulated to state that if K ⊆ L are fields, then K ⊆ L satisfies FIP if and only if there exists an element u ∈ L such that L = K(u) and u is algebraic over K. Indeed, one reason that the FIP concept was introduced by D. D. Anderson, Mullins and the author in  was to seek an algebra-theoretic generalization of the Primitive Element Theorem.
Commutative rings. New research by Lee J. (ed.)