scholarly journals On radicals of infinite matrix rings

1969 ◽  
Vol 16 (3) ◽  
pp. 195-203 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Infinite Matrix ◽  
Natural Numbers ◽  
Index Set ◽  
Matrix Rings ◽  
Infinite Set

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

scholarly journals There are Infinitely Many Fibonacci Primes

2020 ◽  
pp. 1-17
Author(s):  
Fengsui Liu
Keyword(s):  
Residue Class ◽  
Second Order ◽  
Order Topology ◽  
Sieve Method ◽  
Natural Numbers ◽  
Finite Sets ◽  
Index Set ◽  
Infinite Set ◽  
Novel Algorithm

We invent a novel algorithm and solve the Fibonacci prime conjecture by an interaction between proof and algorithm. From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained, then we invent a recursive sieve method, a modulo algorithm on finite sets of natural numbers, for indices of Fibonacci primes. The sifting process mechanically yields a sequence of sets of natural numbers, which converges to the index set of all Fibonacci primes. The corresponding cardinal sequence is strictly increasing. The algorithm reveals a structure of particular order topology of the index set of all Fibonacci primes, then we readily prove that the index set of all Fibonacci primes is an infinite set based on the existing theory of the structure. Some mysteries of primes are hidden in second order arithmetics.

Proceed with care, because some infinities are larger than others

2020 ◽  
pp. 281-292
Author(s):  
Susan D'Agostino
Keyword(s):  
Number Line ◽  
Mathematics Students ◽  
Natural Numbers ◽  
Real Numbers ◽  
Countable Infinity ◽  
One To One ◽  
Infinite Set

“Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

On semiregular infinite matrix rings

1999 ◽  
Vol 27 (12) ◽  
pp. 5737-5740 ◽  
Author(s):  
Francisco José Costa-Cano ◽  
Juan Jacobo simón
Keyword(s):  
Infinite Matrix ◽  
Matrix Rings

Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set

2017 ◽  
Vol 29 (4) ◽  
Author(s):  
Tiwadee Musunthia ◽  
Jörg Koppitz
Keyword(s):  
Natural Numbers ◽  
Maximal Subsemigroups ◽  
Countably Infinite ◽  
Infinite Set

AbstractIn this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

A note on degrees of subsets1

1969 ◽  
Vol 34 (2) ◽  
pp. 256-256 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Minimal Degree ◽  
Turing Degree ◽  
Lower Degree ◽  
Natural Numbers ◽  
Infinite Set

In [2] we constructed an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are nonrecursive sets (e.g. sets of minimal degree) containing no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. We prove that every infinite set must contain a nonrecursive subset of either higher or lower degree.

scholarly journals Generalized cohesiveness

1999 ◽  
Vol 64 (2) ◽  
pp. 489-516 ◽  
Author(s):  
Tamara Hummel ◽  
Carl G. Jockusch
Keyword(s):  
Natural Numbers ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Infinite Set ◽  
Degrees Of Unsolvability ◽  
Computably Enumerable

AbstractWe study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show that for all n ≥ 2, there exists a n-cohesive set. We improve this result for n = 2 by showing that there is a 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0(n+1) and also characterize the jumps of the n-r-cohesive degrees.

Independent Gödel sentences and independent sets

1975 ◽  
Vol 40 (2) ◽  
pp. 159-166
Author(s):  
A. M. Dawes ◽  
J. B. Florence
Keyword(s):  
Positive Integer ◽  
Independent Sets ◽  
Peano Arithmetic ◽  
Natural Numbers ◽  
Simple Set ◽  
Logical Notion ◽  
Infinite Set ◽  
Element Set

In this paper we investigate some of the recursion-theoretic problems which are suggested by the logical notion of independence.A set S of natural numbers will be said to be k-independent (respectively, ∞-independent) if, roughly speaking, in every correct system there is a k-element set (respectively, an infinite set) of independent true sentences of the form x ∈ S. S will be said to be effectively independent (respectively, absolutely independent) if given any correct system we can generate an infinite set of independent (respectively, absolutely independent) true sentences of the form x ∈ S.We prove that(a) S is absolutely independent ⇔S is effectively independent ⇔S is productive;(b) for every positive integer k there is a Π1 set which is k-independent but not (k + 1)-independent;(c) there is a Π1 set which is k-independent for all k but not ∞-independent;(d) there is a co-simple set which is ∞-independent.We also give two new proofs of the theorem of Myhill [1] on the existence of an infinite set of Σ1 sentences which are absolutely independent relative to Peano arithmetic. The first proof uses the existence of an absolutely independent Π1 set of natural numbers, and the second uses a modification of the method of Gödel and Rosser.

Theories incomparable with respect to relative interpretability

1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Natural Numbers ◽  
Infinite Set

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

Hulls of Ring Extensions

2010 ◽  
Vol 53 (4) ◽  
pp. 587-601 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jae Keol Park ◽  
S. Tariq Rizvi
Keyword(s):  
Group Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Affirmative Answer ◽  
Infinite Matrix ◽  
Matrix Rings ◽  
Morita Equivalent ◽  
Semiprime Rings ◽  
Ring Extensions ◽  
The Right

AbstractWe investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.

Sign in / Sign up

Export Citation Format

Share Document