On von Neumann regular rings with an automorphism

2010

A new characterization of von Neumann regular rings is obtained, in terms of simple 0-multiplication of matrices, and is used to establish the natural connections between von Neumann regular rings and feebly Baer modules and rings.

2013

It is well known that to every Boolean ring R can be assigned a Boolean algebra B whose operations are term operations of R. Then a symmetric difference of B together with the meet operation recover the original ring operations of R. The aim of this paper is to show for what a ring R a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular, we reached interesting results if the characteristic of the ring R is either an odd natural number or a power of 2.

International Journal of Mathematics and Mathematical Sciences, 2004

We develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive (bi)sequences of modules over arbitrary commutative ground rings.

Journal of Algebra, 1977

This paper applies various techniques developed in logic to the study of associative rings (not necessarily commutative or with identity). A first-order formula of ring theory is a formula built up in the natural manner using only the logical connectives A (and), v (or), + (implies), 3, V (quantifiers over elements of the ring), the ring theoretic function symbols +, f, 0 and the variables V, , vi ,..., V, (cf. [16]). The theory of a ring R, Th(R), is the set of all first-order sentences (formulas with no free variables) which are true in R. We say R is &categorical if, up to isomorphism, Th(R) has at most one countably infinite model. An introduction to &,-categoric&y in an algebraic setting occurs in [15]. Note that any finite ring is X,-categorical. We have tried to make this paper accessible to anyone familiar with the Wedderburn-Artin structure theory for rings and the rudiments of model theory. Much of our work considers the relationship between the ascending or descending chain conditions usually studied in ring theory and certain ostensibly weaker notions which we define below. We say that the (left) ideal I of a ring R is strictly definable if there is a formula T(Q) of ring theory (with no constants other than 0) such that I is the set of members of R which satisfy v. It is an easy consequence of N,-categoricity that an N,,-categorical ring can have no infinite ascending or descending chain of strictly definable left ideals. In contrast, we show that there is no infinite &,-categorical ring which satisfies either the ascending or descending chain condition on (left) ideals. Thus, Qcategoricity

2000

The aim of this paper is to prove that, if R is a commutative regular ring in which 2 is a unit, then the reduced theory of quadratic forms with invertible coe cients in R, modulo a proper preorder T, satisfies Marshall's signature conjecture and Milnor's Witt ring conjecture (for precise statements, see Section 1 below). For that purpose we

Springer eBooks, 2002

The equivalence problem for deterministic pushdown automata is shown to be decidable. We exhibit a complete formal system for deducing equivalent pairs of deterministic rational boolean series on the alphabet associated with a dpda M. We then extend the result to deterministic pushdown transducers from a free monoid into an abelian group. A general algebraic and logical framework, inspired by Harrison et al.

Communications in Algebra, 2013

2021

We prove model completeness for the theory of addition and the Frobenius map for certain subrings of rational functions in positive characteristic. More precisely: Let p be a prime number, Fp the prime field with p elements, F a field algebraic over Fp and z a variable. We show that the structures of rings R, which are generated over F [z] by adjoining a finite set of inverses of irreducible polynomials of F [z] (e.g., R = Fp[z, 1 z ]), with addition, the Frobenius map x 7→ x and the predicate ‘∈ F ’ together with function symbols and constants that allow building all elements of Fp[z] are model complete, i.e., each formula is equivalent to an existential formula. Further, we show that in these structures all questions, i.e., first order sentences, about the rings R may be, constructively, translated into questions about F .

Houston journal of mathematics

The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prüfer domains and Bezout domains. Let H = {R | R is a commutative ring with 1 = 0 and N il(R) is a divided prime ideal of R}. Let R ∈ H, T (R) be the total quotient ring of R, and set φ : T (R) −→ R N il(R) such that φ(a/b) = a/b for every a ∈ R and b ∈ R \ Z(R). Then φ is a ring homomorphism from T (R) into R N il(R) , and φ restricted to R is also a ring homomorphism from R into R N il(R) given by φ(x) = x/1 for every x ∈ R. A nonnil ideal I of R is said to be φ-invertible if φ(I) is an invertible ideal of φ(R). If every finitely generated nonnil ideal of R is φ-invertible, then we say that R is a φ-Prüfer ring. Also, we say that R is a φ-Bezout ring if φ(I) is a principal ideal of φ(R) for every finitely generated nonnil ideal I of R. We show that the theories of φ-Prüfer and φ-Bezout rings resemble that of Prüfer and Bezout domains.

Journal of Mathematical Sciences, 2003

2010

Journal of Algebra, 2007

We show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to WKL 0 over RCA 0. We also prove that there are computable commutative rings with identity where the nilradical is Σ 0 1-complete, and the Jacobson radical is Π 0 2-complete, respectively.

arXiv: Logic, 2019

We study elementary equivalence of adele rings and decidability for adele rings of general number fields. We prove that elementary equivalence of adele rings implies isomorphism of the adele rings.

arXiv: Logic, 2018

Quantifier elimination of matrix rings $M_n(K)$ for $K$ a formally real field is characterized in the language of rings extended by trace and transposition, in terms of invariant theory. This is used to prove quantifier elimination when $K$ is an intersection of real closed fields. For dimension-free matrices it is shown that no such result can hold by establishing various undecidability results.

Journal of Symbolic Logic, 2002

We use a new version of the Definability Theorem of Beth in order to unify classical Theorems of Yuri Matiyasevich and Jan Denef in one structural statement. We give similar forms for other important definability results from Arithmetic and Number Theory. A.M.S. Classification: Primary 03B99; Secondary 11D99. There is a relation with exponential increment