We investigate necessary and sufficient conditions for aae,f = bb † e,f to hold in rings with involution. Here, ae,f denotes the weighted Moore-Penrose inverse of a, related to invertible and Hermitian elements e, f ∈ R. Thus, some recent results from [7] are extended to the weighted Moore-Penrose inverse.

The article describes Euler’s group Γ (z), formed by the invertible elements of the ring of the residues of the integer complex numbers modulo numbers divisible by z. The answers are proved for the primary cases z = (1+ i)m and z = (p+ iq)m, p2 +q2 = 4k+3 being prime.

$R$ is a unital ring with involution. We investigate the characterizations and representations of weighted core inverse an element in by idempotents units. For example, let $a\in R$ $e\in be invertible Hermitian element, $n\geqslant 1$, then $a$ $e$-core if only there exists (or idempotent) $p$ such that $(ep)^{\ast}=ep$, $pa=0$ $a^{n}+p$ $a^{n}(1-p)+p$) invertible. As consequence, $e, f\in two...

In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of ZZn ⊕ T with no invertible elements, where T is a finite abelian group. We also characterize the lattice ideals that are set-theoretic complete intersections on binomials.

In this paper, we calculate diameters of connected components of commuting graphs of GLn(S), for an integer n ≥ 2 and a commutative antinegative entire semiring S, unless n is a non-prime odd number and S has at least two invertible elements.

The infinite-dimensional analogues of the classical general linear group appear as groups of invertible elements of Banach algebras. Mappings of these groups onto themselves that extend to affine mappings of the ambient Banach algebra are shown to be linear exactly when the Banach algebra is semi-simple. The form of such linear mappings is studied when the Banach algebra is a C*-algebra.

In this paper, diameters of connected components of commuting graphs of GLn(S) are calculated, for an integer n ≥ 2 and a commutative antinegative entire semiring S, unless n is a non-prime odd number and S has at least two invertible elements.

Some inequalities in normed algebras that provides lower and upper bounds for the norm of ∑n j=1 ajxj are obtained. Applications for estimating the quantities ∥∥∥∥x−1∥∥x± ∥∥y−1∥∥ y∥∥ and ∥∥∥∥y−1∥∥x± ∥∥x−1∥∥ y∥∥ for invertible elements x, y in unital normed algebras are also given.

We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We then give a combinatorial classification of reductive monoids by means of the theory of spherical varieties.

Oseledets’ celebrated Multiplicative Ergodic Theorem (MET) [V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210.] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on Rd. When the linear actions are invertible, the MET guarantees an almost-everywhere poi...