## Commutative Rings: New Research by John Lee

By John Lee

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Extra info for Commutative Rings: New Research

Example text

Thus, we have recently found many new examples of domains R such that some maximal chain of overrings of R consists only of integrally closed domains. 3). 8 (c)]. We turn next to the main focus of this section. A ring extension R ⊆ T is said to satisfy FIP (for the “finitely many intermediate rings property”) if there are only finitely many rings D such that R ⊆ D ⊆ T . This concept is relevant because it is obvious that any minimal ring extension satisfies FIP. 2, we note that if R ⊆ T satisfies FIP, then any maximal chain C of R-subalgebras of T must take the form C = {Ri | i = 0, .

Anderson, Mullins and the author in [1] was to seek an algebra-theoretic generalization of the Primitive Element Theorem. 4 presents a complete result along these lines which was found subsequently by G. Picavet, M. Picavet-L’Hermitte, Mullins and the author [13]. 8]. 5]). Let K ⊆ T be a ring extension, where K is a field. Then K ⊆ T satisfies FIP if and only if at least one of the following four conditions holds: (i) K is finite and T is a finite-dimensional K-vector space; (ii) K is infinite, T is a reduced ring, and T = K[α] for some α ∈ T which is algebraic over K; (iii) K is infinite and T = K[α] for some α ∈ T which satisfies α3 = 0; (iv) K is infinite and T = K[α] × K2 × · · · × Kn , where α ∈ T satisfies α3 = 0, and for each i = 2, .

At the beginning of this section, we mentioned that minimal ring extensions could be viewed as playing “the role of the fundamental building blocks in maximal chains” of intermediate rings of ring extensions that satisfy FIP. 5], this fact has been proven when the base ring is an infinite field. 5. Note that this result makes contact with the 22 David E. 4. 10]). Let K be an infinite field and K ⊆ T a ring extension which satisfies FIP. Let T and T be K-subalgebras of T such that T ⊂ T is a minimal ring extension.