# Commutative Rings: New Research by John Lee

By John Lee

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28, 217–219 (2002). [2] Anderson, D. ; Dobbs, D. E. Boll. Un. Mat. 8, 535–545 (2000). [3] Ayache, A. Comm. Algebra 31, 5693–5714 (2003). ; Anderson, D. ; Dobbs, D. E. ; Lecture Notes Pure Appl. Math. 185; Dekker: New York, NY, 1997; 241–250. ; Dobbs, D. ; Lucas, T. G. Rocky Mountain J. , to appear. ; Dobbs, D. E. JP J. , to appear. [7] Davis, E. D. Trans. Amer. Math. Soc. 110, 196–212 (1964). [8] Dech´ene, L. I. Adjacent Extensions Of Rings; Ph. D. thesis; University of California at Riverside: Riverside, CA, 1978.

15. 15 and the subsequent material, we will need the following information and definitions. 4) B/M is isomorphic as an R/M-algebra to exactly one of the following three possibilities: (1) a minimal field extension of R/M, (2) R/M × R/M, (3) (R/M)[X ]/(X 2 ). It will be convenient to say that an integral minimal 26 David E. Dobbs overring B of R (inside K = tq(R)) is of type 1, type 2, or type 3 according as to whether B satisfies condition (1), (2), or (3) above. Given two distinct integral minimal overrings S, T ⊂ K of R, we will say that S, T are a type (a, b) example if S is of type a and T is of type b, while R ⊂ S and R ⊂ T have the same crucial maximal ideal M.

20, which show that behavior of the singly-generated subalgebras is not always predictive of behavior of a given algebra. 10, where one would seek to replace the base ring Z with appropriate base rings R and then classify the faithful unital R-algebras T = R[u] such that R ⊆ T satisfies FIP. 5 by describing the “steps” in any maximal chain of R-subalgebras for a ring extension R ⊆ T that satisfies FIP where the base ring R is not semisimple. We next mention one open question concerning the second theme in Section 2, composites of minimal ring extensions.