Related papers: Rings with finitely many zero divisors
In this article, we prove that in content extentions minimal primes extend to minimal primes and discuss zero-divisors of a content algebra over a ring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We…
In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where…
Let $R$ be a commutative ring with identity $1\neq 0$. In this paper, we continue the study started in [10] concerning when the extended zero-divisor graph of $R$, $\overline{\Gamma}(R)$, is complemented. We also study when…
We establish an explicit inequality for the number of divisors of an integer $n$. It uses the size of $n$ and its number of distinct prime divisors.
We show that for a global field $K$, every ring of $S$-integers has a universal first-order definition in $K$ with $10$ quantifiers. We also give a proof that every finite intersection of valuation rings of $K$ has an existential…
This document seeks to prove there are infinitely many primes whose difference is 2, referred to as twin prime pairs. This proof's methodology involves constructing a function that approximates the number of positive integers, less than a…
We consider the fragment F of first order arithmetic in which quantification is restricted to ''for all but finitely many.'' We show that the integers form an F-elementary substructure of the real numbers. Consequently, the F-theory of…
We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally…
Lower bounds are given for the number of non-real zeros of a second order linear differential polynomial with constant coefficients in a real entire function with finitely many non-real zeros.
A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a…
A ring is rigid if there is no nonzero locally nilpotent derivation on it. In terms of algebraic geometry, a rigid coordinate ring corresponds to an algebraic affine variety which does not allow any nontrivial algebraic additive group…
A finite group is said to have "perfect order classes" if the number of elements of any given order is either zero or a divisor of the order of the group. The purpose of this note is to describe explicitly the finite Hamiltonian groups with…
It is well known that the positive degree cohomology of a finite group G is annihilated by |G|. We improve on this bound in the case of odd degree elements in the integer cohomology ring and show that $e_{odd}(G)$, the exponent of the…
In this paper, we elaborate ring theoretic properties of nodal orders. In particular, we prove that they are closed under taking crossed products with finite groups.
We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…
A $\lambda$-quiddity of size $n$ is an $n$-tuple of elements from a fixed set, which is a solution to a matrix equation that arises in the study of Coxeter's friezes. The study of these solutions involves in particular the use of a notion…
In this paper we provide a complete algebraic characterization of elementary equivalence of rings with a finitely generated additive group in the language of pure rings. The rings considered are arbitrary otherwise.
We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences.
We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under…
We describe all possible ways how a ring can be expressed as the union of three of its proper subrings. This is an analogue for rings of a 1926 theorem of Scorza about groups. We then determine the minimal number of proper subrings of the…