Paul Eakin
University of Kentucky
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Journal of Algebra | 1972
Shreeram S. Abhyankar; William Heinzer; Paul Eakin
If k is a field and X and Y are indeterminates then the statement “consider R = Iz[X, Y] as a polynomial ring in one variable” is ambiguous, for there arc infinitely many possible choices for the ring of coefficients (e.g., If A, = k[X f Y”] then A,[Y] == B,,[Y] 7: R but A,, f J,, if m # n). On the other hand, if Z denotes the integers then the polynomial ring Z[X] has a unique subring over which it is a polynomial ring. This investigation began with our consideration of the first of these examples. In fact, Coleman had asked: If k is a field, then although k[X, 1’1 can be written as a polynomial ring in many different ways, is it true that all of the possible coefficient rings are isomorphic? That is, if T is transcendental over d and 4[T] == k[X, Y], is A a polynomial ring over k ? We found that this is indeed the case (see our (2.8)).We next proved the following: If il is a one dimensional afine domain over a
Journal of Algebra | 1990
Paul Eakin; Avinash Sathaye
eZd and B is a ring such that A [-Xl = B[ E;] zs an equality of polynomial rings, then either A ~ B or there is afield k such that each of A and B is a polynomial +zg in one variabZe over k. This is a corollary of (3.3) in the present paper. Our (7.7) sketches a version of the original proof. In studying this argument, we found that there were implicit in it techniques for investigating the following general question: Suppose A and B are commutative rings with identity and the polynomial rings 4 [X, , . . . , X,,] and B[E; ,..., I’,] are isomorphic, how are A and B related? Are A and B isomorphic? In particular, when does the given isomorphism take i4 onto B ? This study is mainly centered on the latter portion of the question. We are concerned almost entirely with domains It is convenient to use the following terminology which is modeled after that
Journal of Algebra | 1987
Paul Eakin; William Heinzer; Daniel Katz; L.J. Ratliff
where (y,) is a sequence of nonzero elements of F called the sequence of structure constants. Albert called these Cayley-Dickson algebras. The construction, extended to general initial algebras A,, is known as the Cayley-Dickson process [Sl, p. 451 and the algebras A,, that it generates are known as generalized Cayley-Dickson algebras of order n. In the “classical case”, A, is the real numbers and yi = -1 for each i. Then A, is the complex numbers, A1 the quaternions, and A, the Cayley numbers. We are concerned here with the case when A, is a field, F, of characteristic other than 2 or 3, its involution is the identity function, and {y,} is an arbitrary sequence of nonzero elements in F. For n B 3 these Cayley-Dickson algebras are not associative but they do satisfy the flexible law (x(yx) = (xy)x) for each pair x, y E A,, [S2]. This can be stated as saying that the associator (x, y, x) vanishes identically. The strong nonassociutiuity conjecture of P. Yiu [YIU] asserts that the flexible law is the only identity of this type. It says that for x, ZE A,, the associator (x, y, z) vanishes identically if and only if 1, x, and z are linearly dependent over F. We prove a special case of this conjecture (Proposition 1.5) which we then
Canadian Journal of Mathematics | 1968
Paul Eakin; William Heinzer
Let I be an ideal in a Noetherian ring R and let T(I) be the ideal-transform of R with respect to I. Several necessary and sufficient conditions are given for T(I) to be Noetherian for a height one ideal I in an important class of altitude two local domains, and some specific examples are given to show that the integral closure T(I)′ and the complete integral closure T(I)″ of T(I) may differ, even when R is an altitude two Cohen-Macaulay local domain whose integral closure is a regular domain and a finite R-module. It is then shown that T(I)″ is always a Krull ring, and if the integral closure of R is a finite R-module, then T(I)′ is contained in a finite T(I)-module. Finally, these last two results are applied to certain symbolic Rees rings.
Communications in Algebra | 1983
David E. Dobbs; Paul Eakin; Jerome F. Eastham
Let A be an integral domain and K its quotient field. A is called a Krull domain if there is a set { Va) of rank one discrete valuation rings such that A = DaVa and such that each non-zero element of A is a non-unit in only finitely many of the Va. The structure of these rings was first investigated by Krull, who called them endliche discrete Hauptordungen (4 or 5, p. 104). Samuel (7), Bourbaki (1), and Nagata (6) gave an excellent survey of the subject. In terms of the semigroup D(A) of divisors of A, A is a Krull domain if and only if D{A) is an ordered group of the form Z (1, p. 8). In fact, if A is a Krull domain, then the minimal positive elements of D(A) generate D(A) and are in one-to-one correspondence with the minimal prime ideals of A. Moreover, as Bourbaki observed in (1, p. 83), each divisor of A has the form div(Ax + Ay) for some elements x and y of K. In particular, if P is a minimal prime of A, then div(P) = àxv(Ax + Ay); hence P = A : (A : (x,y)). The extent to which the minimal primes of a Krull domain are related to finitely generated ideals has not been completely resolved. This question appeared to be partially answered by Bourbaki in (1, p. 83) when he indicated a method for constructing a two-dimensional Krull ring with a non-finitely generated minimal prime. Our purpose is to show that a domain constructed in the manner suggested by Bourbaki must be noetherian and thus cannot provide the desired example. We make use of the following result of (3) : Let R be a commutative ring with identity and let S be an over ring of R which is a finite unitary R-module. Then if S is noetherian, R is noetherian. Let Z denote the integers and Q the rationals. Define inductively a sequence of algebraic number fields {K^^i such that: (i) Q = Ko, (ii) Ki+i = Ki(yi), where yt is a root of Y 2 — 5at £ Kt[Y], (iii) the integral closure of Z in VJKt is Dedekind. Let A = Z[[X]] and K its quotient field. Set zt= (a tX)K Bourbaki contends that the integral closure of A in i£({s4?=i) is a Krull domain and that the minimal prime generated by X and the zt is not finitely generated. 1 We remark that if the integral closure of Z in L = Q{{ (5a^}°°=1) is
Journal of Algebra | 1979
Stephen McAdam; Paul Eakin
Abstract This paper contributes to the classification of commutative rings having an abundance of Noetherian overrings. Gilmer and Mott have characterized the domains whose proper overrings are Noetherian. In unpublished work, Eakin has shown that adjoining all Noetherian domains to the catalogue of Gilmer-Mott gives a description of the domains whose simple proper overrings are Noetheri an. Using the theory of Krul rings and valuation rings with few zero-divisors, the present article extends the latter characterization to the context of reduced rings
Archive | 1973
Paul Eakin; William Heinzer
Pacific Journal of Mathematics | 1976
Paul Eakin; Avinash Sathaye
Journal of Algebra | 1970
Paul Eakin; William Heinzer
Pacific Journal of Mathematics | 1990
David F. Anderson; David E. Dobbs; Paul Eakin; William Heinzer