A Note on Regular Elements, Inner-Inverses and Summands

A ring R is a Garcia ring provided that the product of two regular elements is unit-regular. We prove that every regular element in a Garcia ring R is the sum/difference of an idempotent and a unit. Furthermore, we prove that every regular element in a weak Garcia ring is the sum of an idempotent and a one-sided unit. These extend several known theorems on (one-sided) unit-regular rings to wider classes of rings with sum summand property.

An element x of a semigroup [ring] A is said to regular if there is an element y of A such that x = xyx, and A is called a regular semigroup [(Von Neumann) regular ring] if every element of A is regular. A quasi-ideal of a semigroup [ring] A is a subsemigroup [subring] Q of A such that AQ intersection QA [is a subset of] Q,and a bi-ideal of A is a subsemingroup [subring] B of A such that BAB [is a subset of] B. Recall that for nonempty subsets X and Y of a ring A, XY denotes the set of all finite sums of the form sigma x[subscript i] y[subscript i] where x[subscript i] [is an element of set] X and Y[subscript i] [is an element of set] Y. We say that a semigroup or a ring has the BQ-property if its quasi-ideals and bi-ideals coincide. It is know that every regular semigroup and every regular ring has the BQ-property. For a nonempty set X, let T(X) denote the full transformation semigroup on X, and for [is an empty set] is not equal to Y[is a subset of] X, let T (X,Y) and T [bar] (X, Y) be the subsemigroups of T(X) defined by T(X, Y)={ alpha is an element of a set T(X) l ran alpha is a subset of Y} and T[bar] (X, Y)={ alpha is an elemaent of a set T(X) l Yalpha is a subset of Y}. Symons and Magill introduced and studied T(X, Y) and T[bar] (X, Y) in 1975 and 1966, respectively. If V is a vector space over a field F, let L[subscript F](V) be the set of all linear transformations alpha : V vector V. For a subspace W of V, define L[subscript F] (V, W) and L[bar][subscript F] (V, W) analogously as follows : L[subscript F] (V, W) = {alpha is an element of a set L[subscriptF(V) l ran alpha is a subset of W} and L[bar][subscript F](V, W) = {alpha is an element of a set L[subscritp F] (V) l { Walpha is a subset of W},and we also consider K[subscript F](V, W) = {alpha L[subscriptF] (V) l W is a subset of ker alpha}. Then L[subscriptF](V, W), L[bar] [subscript F](V, W) and K[subscript F] (V, W) are subsemigroups of (L[subscript F] (V), omicron) and subrings of (L[subscript F] (V),+,omicron) where omicron and+are the composition and usual addition of linear transformations, respectively. This research consists of two major parts. In the first part, we give necessary and sufficient conditions for the elements of these semigroups to be regular. As a consequence, characterizations determinging when these semigroups are regular are given. In the second part, we provide necessary and sufficient conditions for these semigroups and rings to have the BQ-property.

Let be a commutative ring with identity. Two elements and b in are called to be associates if and , or equivalently, if . The generalization of associate relation in R has given the idea for definitions of presimplifiable and weakly presimplifiable rings. First of all, it will be given definitions of very strong associate relation, strong regular associate relation, very strongly associate ring, and strongly regular associate ring. The presimplifiable ring is a commutative ring with the condition that every nonzero element is a unit element. While the weakly presimplifiable ring is a commutative ring with the condition that every nonzero element is regular element. Furthermore, it is shown that the relationship between very strongly associate ring with presimplifiable ring and the linkage between strongly regular associate ring and weakly presimplifiable ring. It is obtained that is a presimplifiable ring if and only if is a very strongly associate ring. Meanwhile, is a weakly presimplifiable ring if and only if is a strongly regular associate ring. Then, it is shown that the correlation between presimplifiable and weakly presimplifiable rings to its polynomial ring and its the formal power series ring . If is a weakly presimplifiable ring, then and are also weakly presimplifiable rings. However, if is a presimplifiable ring, then is also a presimplifiable ring but always not valid for .

Invertible and regular elements in rings with involution

The article describes the experience of applying various approaches to the recognition of some of the most common settlements planning structures, which include radial, regular and ring elements. At the first stage, it is necessary to identify typical properties and elements of the corresponding figures, then find a way to automate the search for such entities in an arbitrary road network of a settlement. To solve the second problem, well-known algorithms are used that are associated with the analysis of the uniformity of the shape of neighborhoods of a settlement. Modifications of the algorithm for searching for radial structural elements are proposed, and the author’s method of detecting ring elements is tested. The selected approaches are implemented in the form of a script in the Python programming language, which sequentially checks the road network given to the input for the presence of lattice or radial-ring planning structure elements. The technique was tested in fifty cities of the world. Verification of the results was carried out by comparing the response of the algorithm with the expert opinion of the research authors or literature on the topic. The accuracy of the classification was 80 %. The algorithm successfully coped with the reference examples of layouts, but experienced difficulties with their modifications, associated, for example, with the lack of closure of the ring elements of the radial-ring structure. The results can be demanded in problems of cartographic generalization, which require recognition and preservation of typical features of spatial objects.

A ring R has weakly stable range one provided that aR+bR=R implies that there exists a <TEX>$y{\in}R$</TEX> such that <TEX>$a+by{\in}R$</TEX> is right or left invertible. We prove, in this paper, that every regular element in an exchange ring having weakly stable range one is the sum of an idempotent and a weak unit. This generalize the corresponding result of one-sided unit-regular ring. Extensions of power comparability and power cancellation are also studied.

In this chapter, we discuss the eigenvalues of certain special elements of pseudo-reflection groups called regular elements. We shall prove that there are significant relations between such eigenvalues and the exponents of the group. Notably, we shall demonstrate that the exponents can be characterized in terms of the eigen-value data of regular elements. The results of this chapter are due to Coxeter [2] and Springer [2].KeywordsWeyl GroupCharacteristic ZeroRegular ElementPrimitive RootReflection GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let T be a weak torsion class of left R-modules and n a positive integer. A left R-module M is called (T, n)-injective if Ext (C, M) = 0 for each (T, n + 1)-presented left R-module C; a right R-module M is called (T, n)-flat if Tor (M, C) = 0 for each (T, n +1)-presented left R-module C; a left R-module M is called (T, n)-projective if Ext (M, N) = 0 for each (T, n)-injective left R-module N; the ring R is called strongly (T, n)-coherent if whenever 0 → K → P → C → 0 is exact, where C is (T, n + 1)-presented and P is finitely generated projective, then K is (T, n)-projective; the ring R is called (T, n)-semihereditary if whenever 0 → K → P → C → 0 is exact, where C is (T, n + 1)-presented and P is finitely generated projective, then pd(K) ⩽ n → 1. Using the concepts of (T, n)-injectivity and (T, n)-flatness of modules, we present some characterizations of strongly (T, n)-coherent rings, (T, n)-semihereditary rings and (T, n)-regular rings.

Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity

The equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X upper A minus upper B upper X equals upper I"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo> − </mml:mo> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>I</mml:mi> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\,XA-BX=I\,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has been well studied in ring theory, operator theory, linear algebra, and other branches of mathematics. In this paper, we show that, in the case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B squared equals upper B"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:msup> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\,B^2=B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the study of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X upper A minus upper B upper X equals upper I"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo> − </mml:mo> <mml:mi>B</mml:mi> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>I</mml:mi> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\,XA-BX=I\,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a noncommutative ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>R</mml:mi> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\,R\,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> leads to several new ways to view and to work with the exchange (or “suitable”) elements in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>R</mml:mi> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\,R\,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the sense of Nicholson. For any exchange element <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A element-of upper R"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>A</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\,A\in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we show that the set of idempotents <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E element-of upper R"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>E</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>R</mml:mi> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\,E\in R\,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E element-of upper R upper A"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>E</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>R</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mi>A</mml:mi> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\,E\in R\,A\,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I minus upper E element-of upper R left-parenthesis upper I minus upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>I</mml:mi> <mml:mo> − </mml:mo> <mml:mi>E</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>R</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>I</mml:mi> <mml:mo> − </mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\,I-E\in R\,(I-A)\,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is naturally parametrized by the roots of a certain left-right symmetric “exchange polynomial” associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\,A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . From the new viewpoints on exchange elements developed in this paper, the classes of clean and strongly clean elements in r

Let R be a ring with involution. It is well-known that an EP element in R is a core invertible element, but the question when a core invertible element is an EP element, the authors answer in this paper. Several new characterizations of star-core, normal and Hermitian elements in R are also presented.

Let ℕ be the set of nonnegative integers and ℤ the ring of integers. Let $$\mathcal{B}$$ be the ring of N × N matrices over ℤ generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of $$\mathcal{B}$$ yields that the subrings generated by them coincide. This subring is the sum of the ideal $$\mathcal{F}$$ consisting of all matrices in $$\mathcal{B}$$ with only a finite number of nonzero entries and the subring of $$\mathcal{B}$$ generated by the identity matrix. Regular elements are also described. We characterize all ideals of $$\mathcal{B}$$ , show that all ideals are finitely generated and that not all ideals of $$\mathcal{B}$$ are principal. Some general ring theoretic properties of $$\mathcal{B}$$ are also established.

On sets of elements in Rickart rings induced by partial orders

We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert calculus, in Advances in Geometry (eds. J.-S. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, Boston, 1995), pp. 147–182, A. Postnikov, On a quantum version of Pieri's formula, in Advances in Geometry (eds. J.-S. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, 1995), pp. 371–383 and A. N. Kirillov and T. Maenor, A Note on Quantum K-Theory of Flag Varieties, preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [E. Mukhin, V. Tarasov and A. Varchenko, Bethe Subalgebras of the Group Algebra of the Symmetric Group, preprint arXiv:1004.4248].Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang–Baxter algebra in a connection with the values of the β-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan–Robbins polytope.

Let M be a finitely generated module of rank r and finite projective dimension over a commutative Noetherian local ring S. We show that M has a free direct summand of rank r if and only if its Buchsbaum–Eisenbud invariant J(M) is a principal ideal. As an application, we consider the problem of finding rigid triples. Recall that the module M is said to be rigid if for each i ≥ 0 and each finitely generated S-module N the condition implies for every j ≥ i. A triple of positive integers b = (b 0, b 1, b 2) is said to be rigid if for every commutative Noetherian local ring S one has that every S-module with minimal free resolution of the form 0 → S b 2 → S b 1 → S b 0 → 0 is rigid. We use our criterion for free direct summands of maximal rank to derive a necessary and sufficient condition for a triple b to be rigid in terms of the rigidity of the universal modules of type b over complete regular local rings. In particular, we exhibit the rigidity of any triple of the form (k, m + 1,m), thus providing a new class of rigid modules.