Boyle’s conjecture and perfect localizations

ABSTRACT

In this article we study the behavior of left QI-rings under perfect localizations. We show that a perfect localization of a left QI-ring is a left QI-ring. We prove that Boyle’s conjecture is true for left QI-rings with finite Gabriel dimension such that every hereditary torsion theory in the Gabriel filtration is perfect. As corollary, we get that Boyle’s conjecture is true for left QI-rings which satisfy the restricted left socle condition, a result proved by Faith in [6 Faith, C. (1976). On hereditary rings and Boyle’s conjecture. Arch. Math. 27(1):113–119.[Crossref] , [Google Scholar]].     

Corrigendum to “reduced tangent cones and conductor at multiplanar isolated singularities”

We explicitly explain how to fix a wrong statement in a preliminary result of our previous paper on the conductor at a multiplanar singularity.     

Prosolvable subgroups of free products of profinite groups

In this paper we establish several equivalent conditions for a commutative ring in which every principal ideal is a finite intersection of prime power ideals to be a general ZPI-ring. Using these results, we establish some equivalent conditions for a commutative ring in which every principal ideal is a finite intersection of primary ideals to be a general ZPI-ring.     

On commutative rings which are strongly prüfer

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.     

GRADE AND GORENSTEIN DIMENSION

Let R be a commutative Noetherian ring and let M be a finite (that is, finitely generated) R-module. The notion grade of M, grade M, has been introduced by Rees as the least integer t ≥ 0 such that Ext ^t _R (M,R) ≠ 0, see [11] Rees, D. 1957. The Grade of an Ideal or Module. Proc. Camb. Phil. Soc., 53: 28–42. [Crossref] , [Google Scholar]. The Gorenstein dimension of M, G-dim M, has been introduced by Auslander as the largest integer t ≥ 0 such that Ext ^t _R (M, R) ≠ 0, see [3] Auslander, M. 1967. Anneaux De Gorenstein Et Torsion En Algebre Commutative Edited by: Mangeney, M., Peskine, C. and Szpiro, L. Paris: Ecole Normale Superieure de Jeunes Filles.  [Google Scholar]. In this paper the R-module M is called G-perfect if grade M = G-dim M. It is a generalization of perfect module. We prove several results for the new concept similar to the classical results.     

The 𝒥-Radical and the Pierce Radicals of a Lattice-Ordered Ring

Abstract

The 𝒥-radical of a lattice-ordered ring Ris the ?-ring analogue of the Jacobson radical of a ring. It is shown that if the matrix ring R _n has the usual lattice order, then 𝒥(R _n ) = 𝒥(R) _n . The connection between an element abeing right ?-quasi-regular and the inequality a ○ x ≤ 0 is also investigated. For squares in an f-ring the connection is an equivalence. In general it is still an equivalence provided xis the sum of elements from a larger f-ring whose absolute values lie in R. It is also shown that the vanishing of annihilators in an f-ring is inherited by enough totally ordered homomorphic images to give a subdirect product decomposition.     

On Prime Rings Satisfying a Certain Generalized Polynomial Identity

Abstract

Recently, Beidar, Fong and Bokut proved that a prime ring satisfies a nontrivial semigroup generalized identity if and only if its central closure is a primitive ring with nonzero socle and the associated skew field is a field. We shall extend their result to the case when generalized polynomial contains three summands.     

Trivial Extensions Defined by Coherent-like Conditions

Abstract

This paper investigates coherent-like conditions and related properties that a trivial extension R ? A ∝ E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition.