Related papers: Rings with finitely many zero divisors

Fields with only finitely many maximal subrings are completely determined. We show that such fields are certain absolutely algebraic fields and give some characterization of them. In particular, we show that the following conditions are…

We show that every polynomial overring of the ring ${\rm Int}(\mathbb Z)$ of polynomials which are integer-valued over $\mathbb Z$ may be considered as the ring of polynomials which are integer-valued over some subset of $\hat{\mathbb{Z}}$,…

We call a group $G$ {\it algorithmically finite} if no algorithm can produce an infinite set of pairwise distinct elements of $G$. We construct examples of recursively presented infinite algorithmically finite groups and study their…

We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem…

A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every…

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in…

The exact degree bound for the generators of rings of polynomial invariants is determined for the finite, non-cyclic groups having a cyclic subgroup of index two. It is proved that the Noether number of these groups equals one half the…

This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…

We prove that in normal rings the tight closure of an ideal can be computed as the sum of the ideal and a piece of the tight closure, called the special tight closure.

We define a closure operation for rings of mixed characteristic and verify that the closure is a ring. We then show that this closure produces a ring with good properties with respect to its Fontaine ring and give an example to show that…

In this work we prove the undecidability (and $\Sigma^0_1$-completeness) of several theories of semirings with fixed points. The generality of our results stems from recursion theoretic methods, namely the technique of effective…

In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that…

Let $R$ be a commutative ring with identity and let $I$ be an ideal of $R$. Let $R\Join I$ be the subring of $R\times R$ consisting of the elements $(r,r+i)$ for $r\in R$ and $i\in I$. We study the diameter and girth of the zero-divisor…

A subgroup of a finite group is wide if each prime divisor of the group order divides the subgroup order. We obtain the description of finite soluble groups with no wide subgroups. We also prove that a finite soluble group with nilpotent…

The finite groups having an indecomposable polynomial invariant whose degree is at least half of the order of the group are classified. Apart from four sporadic exceptions these are exactly the groups having a cyclic subgroup of index at…

We first show a counter intuitive result that in the ring of real valued continuous functions on $[0,1]$ non maximal prime ideals exist. This is a standard proof and a well known result. Interestingly, a non maximal prime ideal in this ring…

We prove that there are infinitely many integers $n$ such that $n$ and $n+1$ have the same number of distinct prime divisors.

We consider in-depth and characterize in certain aspects those rings whose non-units are strongly nil-clean in the sense that they are a sum of commuting nilpotent and idempotent. In addition, we examine those rings in which the non-units…

An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system alone.