we give a new proof of the well known wedderburn's little theorem (1905) that a finite‎ ‎division ring is commutative‎. ‎we apply the concept of frobenius kernel in frobenius representation theorem in finite group‎ ‎theory to build a proof‎.

In this section, R denotes a ring with identity I. We study conditions under which I - Aλ and I - Bλ are left coprime or right coprime, where A, B ∈ R. As applications, we get sufficient conditions under which the Kneading determinant of a finite rank pair of operators on an infinite dimensional space is rational.

We prove that if a positively-graded ring R is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme Tails(R) is a Gorenstein category in the sense of [10]. Moreover, under this condition, a (right) recollement relating Gorensteininjective sheaves in Tails(R) and (graded) Gorenstein-injective R-modules is given.

The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived category of right-modules.

We define topological orthoalgebras (TOAs), and study their properties. While every topological orthomodular lattice is a TOA, the lattice of projections of a Hilbert space (in its norm or strong operator topology) is an example of a lattice-ordered TOA that is not a topological lattice. On the other hand, we show that every compact Boolean TOA is a topological Boolean algebra. We also show tha...

We analyze the transmission and reflection data obtained through transfer matrix calculations on metamaterials of finite lengths, to determine their effective permittivity epsilon and permeability micro. Our study concerns metamaterial structures composed of periodic arrangements of wires, cut wires, split ring resonators (SRRs), closed SRRs, and both wires and SRRs. We find that the SRRs have ...

we investigate buchbaum and eisenbud's construction of the second symmetric power $s_r(x)$ of a chain complex $x$ of modules over a commutative ring $r$. we state and prove a number of results from the folklore of the subject for which we know of no good direct references. we also provide several explicit computations and examples. we use this construction to prove the following ...

Introduction. Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a suppleme...

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.