Earticle

In this article, we compute the order of a general linear group GLn(A) over a finite ring A.

Let R = $Mat_2(F)$ be the ring of all 2 by 2 matrices over a finite field F, X the set of all nonzero, nonunits of R and G the group of all units of R. After investigating some properties of orbits under the left (and right) regular action on X by G, we show that the graph automorphisms group of $\Gamma(R)$ (the zero-divisor graph of R) is isomorphic to the symmetric group $S_{|F|+1}$ of degree |F|+1.

Let R be a commutative ring with identity 1, n ≥ 3, and let 𝒯_n(R) be the linear Lie algebra of all upper triangular n × n matrices over R. A linear map 𝜑 on 𝒯_n(R) is called to be strong commutativity preserving if [𝜑(x), 𝜑(y)] = [x, y] for any x, y ∈ 𝒯_n(R). We show that an invertible linear map 𝜑 preserves strong commutativity on 𝒯_n(R) if and only if it is a composition of an idempotent scalar multiplication, an extremal inner automorphism and a linear map induced by a linear function on 𝒯_n(R).

A simple matrix method to analyze the ship's transverse frame ring is proposed. In this approach, the frame ring is treated as a plane frame of uniform slender members. The loadings on the frame consist of buoyancy loads, deck loads and cargo loads. The hatch coaming are considered to deflect under the loads. Because of symmetry, only the half of the frame is analyzed. The method is to obtain the forces and moments on each member. The deformation of the frame can be determined from the nodal displacements. For a sample calculation, a frame ring of a 10,000 ton class cargo liner is analyzed on the IBM 1130 computer. The numerical results obtained are proved to be resonable.