Regular Ring Research Articles

On rings determined by their idempotents and units

This paper describes properties of three certain classes of rings determined by conditions on idempotents and units, namely, the condition that any two generators of each principal right ideal are associated (UG rings), the condition that every principal right ideal is generated by a sum of a unit and an idempotent (Pr ), and the condition xy = 0 implies xsy = 0 for a sum of idempotent and unit s and any elements x, y of a ring (idun-semicommutative rings). It is proved that the class of all UG rings contains every local as well as every von Neumann regular ring, and the condition Pr is satisfied by both semiperfect and regular rings. Both local and abelian regular rings are proved to be necessarily idun-semicommutative. For all three classes are presented some closure properties and illustrating examples.

The asymptotic Samuel function and invariants of singularities

The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. In this paper, we use this function to introduce the notion of the Samuel slope of a Noetherian local ring, and we study some of its properties. In particular, we focus on the case of a local ring at singular point of a variety, and, among other results, we prove that the Samuel slope of these rings is related to some invariants used in algorithmic resolution of singularities.

ON A THEOREM OF ANDERSON AND CHUN

A commutative ring R is called strongly regular associate if, for any a,b∈R, Ra=Rb implies that a=rb and sa=b for some regular elements r,s∈R. In this paper, we first give a characterization of strongly regular associate rings. A ring R is said to have regular range 1 if, for any a,b∈R, Ra+Rb=R implies that a+bx is a regular for some x∈R. We show that the ring of continuous functions C(X) is strongly regular associate if and only if it has regular range 1. Finally, we generalize a theorem of Anderson and Chun, which states that C([a,b]) is a strongly regular associate ring.

Development and Calculation of an Apparatus with a Ring Tubular Filling for Modernization of a Cooling-Hydration Tower in Thermal Phosphoric Acid Production

Based on the analysis of the work of known designs of enthalpy exchangers, cooling-hydration towers in thermal phosphoric acid production, a design of an enthalpy exchanger with a regular ring tubular filling is developed. As a result of the research, curves of the mass transfer coefficients in the gas phase and the heat transfer coefficients on the gas velocity and irrigation density, the results of calculating the Sherwood and Nusselt numbers on the Reynolds numbers for gas and liquid, are derived. Based on the investigations carried out, an engineering method for calculating the hydrodynamic characteristics, mass transfer coefficients in the gas phase and the heat transfer coefficients, absorption parameters of an apparatus with a ring tubular filling is proposed. This method has been used to calculate the main indicators of the absorption of phosphorus anhydride carried out in the cooling-hydration tower, which has allowed recommending the proposed design of an enthalpy exchanger with a regular ring tubular filling to replace the existing cooling-hydration tower.

On simple generalized m-flat rings

A ring N is known as a right (left) SGmF – ring, if each simple right (left) N -modules are Gm – Flat. In this paper, we investigate characterizations of SGmF – ring. And from the results: 1. If N is a semi-prime, and right SGmF–ring, then the C(N) is regular ring. 2. If N is SGmF – ring, with every ideal of N is principal then: a. N is NJ iff N is feckly reduced, b. N is m – regular ring, if N is ZC ring.

On the structure of certain von Neumann regular and π-regular rings with prime center

On the structure of certain von Neumann regular and π-regular rings with prime center

On Pythagorean fuzzy ideals of a classical ring

<abstract><p>The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set and is an effective approach of handling uncertain situations. Ring theory is a prominent branch of abstract algebra, vibrant in wide areas of current research in mathematics, computer science and mathematical/theoretical physics. In the theory of rings, the study of ideals is significant in many ways. Keeping in mind the importance of ring theory and Pythagorean fuzzy set, in the present article, we characterize the concept of Pythagorean fuzzy ideals in classical rings and study its numerous algebraic properties. We define the concept of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal and prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms a ring under certain binary operations. Furthermore, we present Pythagorean fuzzy version of the fundamental theorem of ring homomorphism. We also introduce the concept of Pythagorean fuzzy semi-prime ideals and give a detailed exposition of its different algebraic characteristics. In the end, we characterized regular rings by virtue of Pythagorean fuzzy ideals.</p></abstract>

A note on the Moore-Penrose inverse of block matrices

Motivated by the representation for the Moore-Penrose inverse of the block matrix over a *-regular ring presented in [R.E. Hartwig and P. Patr?cio, When does the Moore-Penrose inverse flip? Operators and Matrices, 6(1):181-192, 2012], we show that the formula of the Moore-Penrose inverse is the same as the expression given by [Nieves Castro-Gonz?lez, Jianlong Chen and Long Wang, Further results on generalized inverses in rings with involution, Elect. J. Linear Algebra, 30:118-134, 2015].

Flat modules and coherent endomorphism rings relative to some matrices

<abstract><p>Let $ N $ be a left $ R $-module with the endomorphism ring $ S = \text{End}(_{R}N) $. Given two cardinal numbers $ \alpha $ and $ \beta $ and a matrix $ A\in S^{\beta\times\alpha} $, $ N $ is called flat relative to $ A $ in case, for each $ x\in l_{N^{(\beta)}}(A) = \{u\in N^{(\beta)} \mid uA = 0\} $, there are a positive integer $ k $, $ y\in N^{k} $ and a $ k\times \beta $ row-finite matrix $ C $ over $ S $ such that $ CA = 0 $ and $ x = yC $. It is shown that $ N_{S} $ is flat relative to a matrix $ A $ if and only if $ l_{N^{(\beta)}}(A) $ is generated by $ N $. $ S $ is called left coherent relative to $ A $ if Ker$ (_{S}S^{(\beta)}\to _{S}S^{(\beta)}A) $ is finitely generated. It is shown that $ S $ is left coherent relative to $ A $ if and only if Hom$ _{R}(N, l_{N^{n}}(A)) $ is a finitely generated left $ S $-module if and only if $ l_{N^{n}}(A) $ has an add$ (N) $-precover (add$ (N) $ denotes the category of all direct summands of finite direct sums of copies of $ _{R}N $). Regarding applications, new necessary and sufficient conditions for epic (monic, having the unique mapping property) add$ (N) $-precovers of $ l_{N^{(\beta)}}(A) $ are investigated. Also, some new characterizations of left $ n $-semihereditary rings and von Neumann regular rings are given.</p></abstract>

Lower bounds on Hilbert–Kunz multiplicities and maximal F-signatures

AbstractHilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunz multiplicity.

Unimodular rows over monoid extensions of overrings of polynomial rings

Let R be a commutative Noetherian ring of dimension d and M a commutative cancellative torsion-free seminormal monoid. Then: (1) Let A be a ring of type R[d,m,n] and P be a projective A[M]-module of rank r≥max {2,d+1}. Then the action of E(A[M]⊕P) on (A[M]⊕P) is transitive and (2) Assume (R,m,K) is a regular local ring containing a field k such that either char k=0 or char k=p and tr-deg K∕𝔽p≥1. Let A be a ring of type R[d,m,n]∗ and f∈R be a regular parameter. Then all finitely generated projective modules over A[M], A[M]f and A[M]⊗RR(T) are free. When M is free both results are due to Keshari and Lokhande (2014).

Realizing ultragraph Leavitt path algebras as Steinberg algebras

In this article, we realize the finite range ultragraph Leavitt path algebras as Steinberg algebras. This realization allows us to use the groupoid approach to obtain structural results about these algebras. Using the skew product of groupoids, we show that ultragraph Leavitt path algebras are graded von Neumann regular rings. We characterize strongly graded ultragraph Leavitt path algebras and show that every ultragraph Leavitt path algebra is semiprimitive. Moreover, we characterize irreducible representations of ultragraph Leavitt path algebras. We also show that ultragraph Leavitt path algebras can be realized as Cuntz-Pimsner rings.

Drazin inverse and generalization of core-nilpotent decomposition

The Drazin inverse is connected with the notion of index and core-nilpotent decomposition whenever it is discussed in the context of ring of matrices over complex field. In the absence of Drazin inverse for a given element from an arbitrary associative ring (not necessarily with unity), in this paper, the notion of right (left) core-nilpotent decomposition has been introduced and established its relations with right (left) [Formula: see text]-regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) [Formula: see text]-regular if and only if it has a right (left) core-nilpotent decomposition. In the process, several properties of sharp order in an associative ring are studied and with the help of the same, new characterizations of Drazin inverse over an associative ring are obtained and the relation between core-nilpotent decomposition and the Drazin inverse is obtained.

On trace zero matrices and commutators

Given any commutative ring R, a commutator of two n×n matrices over R has trace 0. In this paper, we study the converse: whether every n×n trace 0 matrix is a commutator. We show that if R is a regular ring with large enough Krull dimension relative to n, then there exists a n×n trace 0 matrix that is not a commutator. This improves on a result of Lissner by increasing the size of the matrix allowed for a fixed R. We also give an example of a Noetherian dimension 1 commutative domain R that admits a n×n trace 0 non-commutator for any n≥2.

On the ideal transforms defined by an ideal

Let R be a commutative Noetherian ring, I an ideal of R and M an R-module. The ambiguous structure of I-transform functor DI (−) makes the study of its properties attractive. In this paper we gather conditions under which, DI (R) and DI (M) appear in certain roles. It is shown, under these conditions that DI (R) is a Cohen-Macaulay ring, regular ring, · · · and DI (M) can be regarded as a Noetherian, flat, · · ·R-module. We also present a primary decomposition of zero submodule of DI (M) and secondary representation of DI (M).

Twists of rational Cherednik algebras

Abstract We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one.

Skew McCoy rings and σ-compatibility

In this paper, we study [Formula: see text]-skew McCoy rings under the [Formula: see text]-compatible or the [Formula: see text]-semicompatible conditions. We show that if [Formula: see text] is a semicommutative right or left artinian ring which is [Formula: see text]-semicompatible with an epimorphism [Formula: see text], then the Jacobson radical [Formula: see text] is [Formula: see text]-skew McCoy. As a corollary, we get that the Jacobson radical of a semicommutative artinian ring is right McCoy. We also show that every [Formula: see text]-compatible right duo ring is [Formula: see text]-skew McCoy and that for [Formula: see text]-compatible regular rings, the notions of the [Formula: see text]-skew McCoy and the right McCoy coincide. In addition, we show that every [Formula: see text]-semicompatible semicommutative ring is linearly [Formula: see text]-skew Camillo and that every matrix ring over a division ring is linearly [Formula: see text]-skew Camillo for any endomorphism [Formula: see text].

Vanishing of Avramov obstructions for products of sequentially transverse ideals

Two ideals I and J are called transverse if I ∩ J = I J . We show that the obstructions defined by Avramov for products of (sequentially) transverse ideals in regular local rings are always 0. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative multiplicative structures on the minimal free resolution of the quotient defined by products of transverse ideals.

Tate algebras and Frobenius non-splitting of excellent regular rings

An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that excellent F -pure rings of prime characteristic are not Frobenius split in general, even for Euclidean domains. Our construction uses the existence of a complete non-Archimedean field k of characteristic p with no nonzero continuous k -linear maps k^{1/p} \to k . An explicit example of such a field is given based on ideas of Gabber, and may be of independent interest. Our examples settle a long-standing open question in the theory of F -singularities whose origin can be traced back to when Hochster and Roberts introduced the notion of F -purity. The excellent Euclidean domains we construct also admit no nonzero R -linear maps R^{1/p} \rightarrow R . These are the first examples that illustrate that F -purity and Frobenius splitting define different classes of singularities for excellent domains, and are also the first examples of excellent domains with no nonzero p^{-1} -linear maps. The latter is particularly interesting from the perspective of the theory of test ideals.

Local Cohomology of Module of Differentials of integral extensions II

In this note (R,m) denotes a complete regular local ring and B mostly denotes its absolute integral closure. The four objectives of this paper are the following: i) to determine the highest non-vanishing local cohomology of ΩB/R in equicharacteristic 0, ii) to establish a connection between each of ΩB/R and ΩB/V and pull-back of ΩA/V via a short exact sequence together with new observations on corresponding local cohomologies in mixed characteristic where V is the coefficient ring of R and A is its absolute integral closure, iii) to demonstrate that ΩB/R can be mapped onto a cohomologically Cohen-Macaulay module and iv) to study torsion-free property for ΩC/V and ΩC/k along with their respective completions where C is an integral domain and a module finite extension of R. In this connection an extension of Suzuki's theorem on normality of complete intersections to the formal set-up in all characteristics is accomplished.