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AbstractLet R be a commutative ring with identity and R′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R′ is finitely generated, i.e., R′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R≤2 , then R′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R=∞ or reg-dim R=n≤ dim R=n+m−1 for arbitrary positive integers n, m. [ABSTRACT FROM AUTHOR]

By means of the maximal left ring of quotients, this paper proves that every n-Lie m-derivation on a triangular ring can be resolved into a sum of an extremal n-derivation and an n-additive central mapping. As applications, n-Lie m-derivations on some classical triangular rings are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we give a necessary and sufficient condition for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroup using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also discuss nice extensions for simplicial affine semigroups, a concept which is motivated by nice extensions of numerical semigroups. [ABSTRACT FROM AUTHOR]

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A (Δ) = k [ x 1 , ... , x n ] / 〈 x 1 2 , ... , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A (Δ) in terms of the simplicial complex Δ. We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A (Δ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization. [ABSTRACT FROM AUTHOR]

The famous Kurosh Problem for division rings asks whether a finitely generated algebraic division ring is centrally finite. This longstanding problem posed in 1941 by Kurosh remains without answer up to now. In this paper, we study some questions related to this problem for some classes of division rings. [ABSTRACT FROM AUTHOR]

For Abelian p-groups, Goldsmith, Salce, et al., introduced the notion of minimal full inertia. In parallel to this, we define the concept of minimal characteristic inertia and explore those p-primary Abelian groups having minimal characteristic inertia. We establish the surprising result that, for each Abelian p-group A, the square A ⊕ A has the minimal characteristic inertia if, and only if, it has the minimal full inertia. We also obtain some other relationships between these two properties. Specifically, we exhibit groups which do not have neither of the properties, as well as we show via a concrete complicated construction by using some nonstandard tricks from ring theory and module theory that, for any prime p, there is a p-group possessing the minimal characteristic inertia which does not possess the minimal full inertia. [ABSTRACT FROM AUTHOR]

Recall that a commutative ring R is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module M is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules. [ABSTRACT FROM AUTHOR]

We study the ring of polyfunctions over Z / n Z. The ring of polyfunctions over a commutative ring R with unit element is the ring of functions f : R → R which admit a polynomial representative p ∈ R [ x ] in the sense that f (x) = p (x) for all x ∈ R. This allows to define a ring invariant s which associates to a commutative ring R with unit element a value in N ∪ { ∞ }. The function s generalizes the number theoretic Smarandache function. For the ring R = Z / n Z we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number Ψ (n) of polyfunctions over Z / n Z. We also investigate algebraic properties of the ring of polyfunctions over Z / n Z. In particular, we identify the additive subgroup of the ring and the ring structure itself. Moreover we derive formulas for the size of the ring of polyfunctions in several variables over Z / n Z , and we compute the number of polyfunctions which are units of the ring. [ABSTRACT FROM AUTHOR]

The zero-divisor graphs of commutative rings have been used to build bridges between ring theory and graph theory. Namely, they have been used to characterize many ring properties in terms of graphic ones. However, many results are established only for reduced rings because a zero-divisor graph defined in the classical manner lacks the information on relationship between powers of zero-divisors. The aim of this article is to remedy this situation by introducing a parametrized family of graphs { Γ ¯ i (R) } i ∈ N * , for a ring R, which reveals more of the relationship between powers of zero-divisors as follows: For each i ∈ N * , Γ ¯ i (R) is the simple graph whose vertex set is the set of non-zero zero-divisors such that two distinct vertices x and y are joined by an edge if there exist two positive integers n ≤ i and m ≤ i such that x n y m = 0 with x n ≠ 0 and y m ≠ 0. Our aim is to study in detail the behavior of the filtration { Γ ¯ i (R) } i ∈ N * as well as the relations between its terms. We give answers to several interesting and natural questions that arise in this context. In particular, we characterize girth and diameter of Γ ¯ i (R) and give various examples. [ABSTRACT FROM AUTHOR]