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MR0044509 (13,427c) 09.0X Artin, Emil; Tate, John T. A note on finite ring extensions. J. Math. Soc. Japan 3 (1951), 74–77. Let R and S be two commutative rings, R ⊆ S. Then S is called a modulefinite extension of R, if it is an R-module with a finite set of generators, that is, if there exists a finite number of elements of S such that every element of S can be represented as a linear combination of them with coefficients in R. On the other hand, S is said to be a ring-finite extension of R, if it can be written in the form S = R[ξ1, ξ2, · · · , ξn]. The following theorem is proved. Let R be a Noetherian ring with unit element, let S be a ring-finite extension, and let T be an intermediate ring, R ⊆ T ⊆ S, such that S is a module-finite extension of T . Then T is a ring-finite extension of R. As an application, the following theorem of Zariski [Bull. Amer. Math. Soc. 53, 362–368 (1947); MR0020075] is obtained. If a ring-finite extension of a field is a field, then it is algebraic and hence module-finite. As shown by Zariski, the Nullstellensatz is an immediate consequence of this result. It is further shown that a Noetherian integral domain R with a unit element has ring-finite extensions which are fields if and only if the quotient field F of R is a ring-finite extension of R. The ring-finite extension fields of R are then exactly the module-finite extension fields of F . The condition that F is a ring-finite extension of R is equivalent to each of the following conditions. I. There exists an element a = 0 of R which is contained in all proper prime ideals of R. II. There exists only a finite number of minimal prime ideals of R. III. There exists only a finite number of prime ideals of R, and every one of them is maximal. R. Brauer From MathSciNet, June 2017