Non-Commutative Ring Theory Research Papers (original) (raw)

We use the concept of 2-absorbing ideal introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local... more

We use the concept of 2-absorbing ideal introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local TAF-domains are the atomic pseudo-valuation domains.

Let G be a group and R be a G-graded commutative ring, i.e., R = g2G Rg and RgRhRgh for all g, h2G. In this paper, we study the graded primary ideals and graded primary G-decomposition of a graded ideal.

We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one... more

We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory.

The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce... more

The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Lambda-algebraic geometry. We show that Lambda-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.

In this paper we introduce an algebra embedding ι:K< X >→ S from the free associative algebra K< X > generated by a finite or countable set X into the skew monoid ring S = P * Σ defined by the commutative polynomial ring P =... more

In this paper we introduce an algebra embedding ι:K< X >→ S from the free associative algebra K< X > generated by a finite or countable set X into the skew monoid ring S = P * Σ defined by the commutative polynomial ring P = K[X× N^*] and by the monoid Σ = < σ > generated by a suitable endomorphism σ:P→ P. If P = K[X] is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra S = ⊕_i S_i with S_i = P σ^i and σ:P → P an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of P. Moreover, using a suitable grading for the algebra P compatible with the action of Σ, we obtain a bijective correspondence, preserving Gröbner bases, between graded Σ-invariant ideals of P and a class of graded two-sided ideals of S. By means of the embedding ι this results in the unification, in the graded case, of the Gröbner bases ...

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example (1), (2), (3), (4), (5) and (8)) asserting that the existence of a... more

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example (1), (2), (3), (4), (5) and (8)) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands.... more

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.

Let AAA be a ring and MA\M_AMA the category of AAA-modules. It is well known in module theory that for any AA A-bimodule BBB, BBB is an AAA-ring if and only if the functor −otimesAB:MAtoMA-\otimes_A B: \M_A\to \M_A−otimesAB:MAtoMA is a monad (or triple). Similarly, an... more

Let AAA be a ring and MA\M_AMA the category of AAA-modules. It is well known in module theory that for any AA A-bimodule BBB, BBB is an AAA-ring if and only if the functor −otimesAB:MAtoMA-\otimes_A B: \M_A\to \M_A−otimesAB:MAtoMA is a monad (or triple). Similarly, an AA A-bimodule C\CC is an AAA-coring provided the functor −otimesAC:MAtoMA-\otimes_A\C:\M_A\to \M_A−otimesAC:MAtoMA is a comonad (or

An ideal of a ring is completely irreducible if it is not the intersec- tion of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without niteness conditions. It is known... more

An ideal of a ring is completely irreducible if it is not the intersec- tion of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without niteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that

In this paper, we extend the characterizations of Kuroki [17], by initiating the concept of intuitionistic fuzzy left (resp. right, interior, quasi-, bi-, generalized bi-) ideals in a class of non-associative and non-commutative rings... more

In this paper, we extend the characterizations of Kuroki [17], by initiating the concept of intuitionistic fuzzy left (resp. right, interior, quasi-, bi-, generalized bi-) ideals in a class of non-associative and non-commutative rings (LA-ring). We characterize regular (intra-regular, both regular and intra-regular) LA-rings in terms such ideals.

Abstract: Let $ R $ be a unital algebra over reals and $ K\ subseteq Hom (R,\ mathbb {R}) ,closedwithrespecttotheproducttopology.Weconsider, closed with respect to the product topology. We consider ,closedwithrespecttotheproducttopology.Weconsider R $ endowed with the topology induced by the family of seminorms $\ rho_ {\... more

Abstract: Let $ R $ be a unital algebra over reals and $ K\ subseteq Hom (R,\ mathbb {R}) ,closedwithrespecttotheproducttopology.Weconsider, closed with respect to the product topology. We consider ,closedwithrespecttotheproducttopology.Weconsider R $ endowed with the topology induced by the family of seminorms rhoalpha(a):=∣alpha(a)∣\ rho_ {\ alpha}(a):=|\ alpha (a)| rhoalpha(a):=∣alpha(a)∣, for alphainK\ alpha\ in K alphainK and $ a\ in R .Incase. In case .Incase K $ is compact, we also consider the topology induced by ∣a∣K:=supalphainK∣alpha(a)∣\| a\| _K:=\ sup_ {\ alpha\ in K}|\ alpha (a)| ∣a∣_K:=sup_alphainK∣alpha(a)∣ for $ a\ in R .If. If .If K $ is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d- ...

Ž . Journal of Algebra 217, 434447 1999 Article ID jabr.1998.7840, available online at http:rrwww.idealibrary.com on ... The Zero-Divisor Graph of a Commutative Ring ... David F. Anderson and Philip S. Livingston ... Mathematics... more

Ž . Journal of Algebra 217, 434447 1999 Article ID jabr.1998.7840, available online at http:rrwww.idealibrary.com on ... The Zero-Divisor Graph of a Commutative Ring ... David F. Anderson and Philip S. Livingston ... Mathematics Department, The Uni¨ersity of ...

We exhibit new examples of double quasi-Poisson brackets, based on some classification results and the method of fusion. This method was introduced by Van den Bergh for a large class of double quasi-Poisson brackets which are said... more

We exhibit new examples of double quasi-Poisson brackets, based on some classification results and the method of fusion. This method was introduced by Van den Bergh for a large class of double quasi-Poisson brackets which are said differential, and our main result is that it can be extended to arbitrary double quasi-Poisson brackets. We also provide an alternative construction for the double quasi-Poisson brackets of Van den Bergh associated to quivers, and of Massuyeau-Turaev associated to the fundamental groups of surfaces.

In this survey article we outline the history of the twin theories of weak normality and seminormality for commutative rings and algebraic varieties with an emphasis on the recent developments in these theories over the past fifteen... more

In this survey article we outline the history of the twin theories of weak normality and seminormality for commutative rings and algebraic varieties with an emphasis on the recent developments in these theories over the past fifteen years. We develop the theories for general commutative rings, but specialize to reduced Noetherian rings when necessary. We hope to acquaint the reader

We consider the abelian group PTPTPT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of... more

We consider the abelian group PTPTPT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of "multiplication" bullet\bulletbullet on the collection of DG categories which makes this abelian group into a commutative ring. A few applications are considered: representability of "standard" functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.

A number of main properties of the commuting regular rings and commuting regular semigroups have been studied in this paper. Some significant results of which will be used for the commutative rings and a necessary and sufficient condition... more

A number of main properties of the commuting regular rings and commuting regular semigroups have been studied in this paper. Some significant results of which will be used for the commutative rings and a necessary and sufficient condition is given for a semigroup to be commuting regular.