It is well-known that an element of a commutative ring with identity is nilpotent if, and only if , it lies in every prime ideal of the ring. A modification of this fact is amenable to a very simple proof mining analysis. We formulate a quantitative version of this modification and obtain an explicit bound. We present an application. This proof mining analysis is the leitmotif for some comments and observations on the methodology of computational extraction. In particular, we emphasize that the formulation of quantitative versions of ordinary mathematical theorems is of independent interest from proof mining metatheorems.

Let $R$ be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on $R$, $\Gamma \left( R \right)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra\,+\,Rb\,=\,R$. In this paper, we consider a subgraph ${{\Gamma }_{2}}\left( R \right)$ of $\Gamma \left( R \right)$ that consists of non-unit elements. We investigate the behavior of ${{\Gamma }_{2}}\left( R \right)$ and ${{\Gamma }_{2}}\left( R \right)\backslash \text{J}\left( R \right)$, where $\text{J}\left( R \right)$ is the Jacobson radical of $R$. We associate the ring properties of $R$, the graph properties of ${{\Gamma }_{2}}\left( R \right)$, and the topological properties of $\text{Max}\left( R \right)$. Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.

We introduce and investigate the notion of envelope dimension of commutative rings and modules over them. In particular, we show that the envelope dimension of a ring, $R$, is equal to that of the $R$-module ${{\mathbb{R}}^{\left( \mathbb{N} \right)}}$. We also prove that the Krull dimension of a ring is no more than its envelope dimension and characterize Noetherian rings for which these two dimensions are equal. Moreover, we generalize and study the concept of simplified radical formula for modules, which we defined in an earlier paper.

Let $R$ be a commutative ring with identity, and let $M$ be a unitary module over $R$. We call $M$$\text{H}$-smaller ($\text{HS}$ for short) if and only if $M$ is infinite and $\left| M/N \right|\,<\,\,\left| M \right|$ for every nonzero submodule $N$ of $M$. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: suppose $M$ is faithful over $R$, $R$ is a domain (we will show that we can restrict to this case without loss of generality), and $K$ is the quotient field of $R$. If $M$ is $\text{HS}$ over $R$, then $R$ is $\text{HS}$ as a module over itself, $R\,\subseteq \,M\,\subseteq \,K$, and there exists a generating set $S$ for $M$ over $R$ with $\left| S \right|\,<\,\left| R \right|$. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.

The zero-divisor graph $\Gamma (R)$ of a commutative ring $R$ is the graph whose vertices consist of the nonzero zero-divisors of $R$ such that distinct vertices $x$ and $y$ are adjacent if and only if $xy\,=\,0$. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings $R$ such that $\Gamma (R)$ is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

This paper is a continuation for the study of the zero-divisor graph for the ring of Gaussian integers modulo n, Γ(ℤn[i]) in [8] (Emad Abu Osba, Salah Al-Addasi and Nafez Abu Jaradeh. Zero divisor graph for the ring of Gaussin integers modulo n. Comm. Algebra 36(10) (2008), 3865–3877). It is investigated, when is Γ(ℤn[i]) locally H, Hamiltonian or bipartite graph? A full characterisation for the chromatic number and the radius is also given.

In this paper, we prove that any Zinbiel algebra can be endowed with the structure of commutative algebra with divided powers. We introduce the notion of universal enveloping Zinbiel algebra of a commutative algebra with divided powers algebras. We prove that the free divided powers algebra on a free module M, is the divided powers sub-algebra generated by M, of the divided powers algebra induced by the free Zinbiel algebra on M. Finally, we construct a basis for the enveloping Zinbiel algebra.

A ring is called clean if every element is the sum of an idempotent and a unit. It is an open question whether the tensor products of two clean algebras over a field is clean. In this note we study the tensor product of clean algebras over a field and we provide some examples to show that the tensor product of two clean algebras over a field need not be clean.

Eventual positivity problems for real convergent Maclaurin series lead to density questions for sets of harmonic functions. These are solved for large classes of series, and in so doing, asymptotic estimates are obtained for the values of the series near the radius of convergence and for the coefficients of convolution powers.

The Chinese Remainder Theorem states that if I and J are comaximal ideals of a ring R, then A/(I∩J)A is isomorphic to A/IA×A/JA for any left R-module A. In this paper we study the converse; when does A/(I∩J)A and A/IA×A/JA isomorphic imply that I and J are comaximal?