Showing papers on "Ring of integers published in 1971"

Diophantine Representation of Recursively Enumerable Predicates

Abstract: Publisher Summary This chapter presents a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients that devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. It is well-known that an algorithm for determining the solvability in integers would yield an algorithm for determining the solvability in positive integers and conversely. Hence, the solvability in positive integers is discussed. Lower-case Latin letters will always be variables whose range is the positive integers. Every recursively enumerable set of positive integers (e.g., the set of all prime numbers) coincides with the set of all positive values of some polynomial with integer coefficients.

Almost Everywhere Convergence of Fourier Series on the Ring of Integers of a Local Field

Abstract: It is shown that the partial sums of the Fourier series of $L^p (\mathfrak{D})$-functions $(p > 1)$ converge almost everywhere (a.e.), where $\mathfrak{D}$ is the ring of integers in a local field K. This includes the case where K is a p-adic number field as well as the case where $\mathfrak{D}$ is the Walsh–Paley or dyadic group $2^\omega $. The techniques are essentially those used by Carleson [2] in establishing the a.e. convergence of trigonometric Fourier series for $L^2 ( - \pi ,\pi )$-functions as modified by Hunt [4] to obtain this same result for $L^p ( - \pi ,\pi )$-functions, $p > 1$. The necessary modifications for the local field setting are made in the context of the Sally’Taibleson [7] development of harmonic analysis on local fields and by use of Taibleson’s multiplier theorem [11]. These same results for $2^\omega $ have already been obtained by Billiard $(L^2 (2^\omega ))$ [1] and by Sjolin $(L^p (2^\omega ))$, $p > 1$) [8]. Many advantages (in particular the non-Archimidean nature of th...

Characterization of Some Fully Ordered Rings

Abstract: In a fully ordered (f.o.) ring with identity, the set of all bounded elements as defined below might be an Archimedian subring. Most of the examples of f.o. rings constructed in literature having the bounded set as Archimedian subring are polynomial rings. For example I[x] , R[x] etc., where I is the ring of integers and R is the field of rationals, with lexicographic ordering. Now we ask whether a f.o. ring with identity, with the set of bounded elements as Archimedian subring can be a polynomial ring over an Archimedian subring. This is answered affirmatively in Theorem 1. It is proved in Theorem 3 that f.o. rings with identity and with every positive element a large element, belong to the above class. The problem then arises as to when the set of all bounded elements, called a weak Archimedian sub- ring in [2], becomes an Archimedian subring. This problem is completely solved in Theorem 2. The concept of weak Archimedian rings is found to be useful by the author in characterizing some f.o. rings as algebraic algebras in [3].

Rings with the contraction property

Abstract: A ring R (not necessarily commutative or with unit) has the contraction property iff every ideal of every subring of R is a contracted ideal. It is shown that R is a primitive ring with the contraction property iff R is an absolutely algebraic field. This result, together with the fact that the Jacobson Radical of a ring with the contraction property is nil, shows that a nil semisimple ring with the contraction property is a subdirect sum of absolutely algebraic fields (and is therefore commutative). It is shown that if R is a torsion free nil ring with the contraction property then R2 = (0). It follows that any torsion free ring with the contraction property is the extension of a zero ring and a subdirect sum of absolutely algebraic fields. Also, if R is a nil ring with the contraction property then R2 is torsion as an additive group. 1. Rings with the contraction property. DEFINITION 1.1. A ring R (not necessarily commutative or with unit) has the contraction property iff every ideal of every subring of R is a contracted ideal; that is, if S is a subring of R and I an ideal of S, we have I=TfliS for some ideal T of R [6, pp. 218-221]). Hereafter, we call R a c-ring iff it satisfies the condition of Definition 1.1. The ring of integers is a simple example of a c-ring. Another example is any absolutely algebraic field F [3, p. 147], for if F is any absolutely algebraic field and S is any subring of F, it is easy to check that S is actually a subfield of F. Thus, the only ideals of S are (0) and S which are contracted ideals. THEOREM 1.1. The class of all c-rings is closed under homomorphisms, direct sums, if each summand is a ring with unit, and is hereditary for subrings. PROOF. (1) Let A be a c-ring and B a homomorphic image of A via a homomorphism (p. Let S be a subring of B, I an ideal of S. (po1 (I) is an ideal in po-l(S), a subring of A, thus so-l(I) = T \p-'(S), with Received by the editors February 16, 1970 and, in revised form, April 17, 1970. AMS 1969 subject classifications. Primary 1620, 1932, 1642.

On the representation of positive integers as sums of three cubes of positive rational numbers

Abstract: A one parameter solution of th equation was given by Ryley in 1825. Others were found by Richmond and myself. In the Landau memorial volume [1] recently published, there appears a joint paper, with the above title, of Davenport and Landau dating from 1935. They prove that if n is a positive integer, then positive rational solutions of (1) exist with denominators of order of magnitude O ( n 2 ). Their proof depended upon a two parameter solution of (1) due to Richmond and is very complicated.