Narrow Results

In this paper, we study the relation between the generalized inverse properties of an element in a ring with involution and related equations. Mainly, by exploring the existence of the solution in a given set and the expressions of the general solution to these constructed related equation, we obtain a lot of new characterizations of EP elements, partial isometries, SEP elements, Hermitian elements and normal elements.

By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group SL_n(ℤ_p) is a unit.

Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let Λ_n = K ⌊x₁, …, x_n⌋ be the Grassmann algebra over a commutative ring K with ½ ∈ K, and δ be a skew K-derivation of Λ_n. It is proved that δ is a unique sum δ = δ^ev + δ^od of an even and odd skew derivation. Explicit formulae are given for δ^ev and δ^od via the elements δ (x₁), …, δ (x_n). It is proved that the set of all even skew derivations of Λ_n coincides with the set of all the inner skew derivations. Similar results are proved for derivations of Λ_n. In particular, Der_K(Λ_n) is a faithful but not simple Aut_K(Λ_n)-module (where K is reduced and n ≥ 2). All differential and skew differential ideals of Λ_n are found. It is proved that the set of generic normal elements of Λ_n that are not units forms a single Aut_K(Λ_n)-orbit (namely, Aut_K(Λ_n)x₁) if n is even and two orbits (namely, Aut_K(Λ_n)x₁ and Aut_K(Λ_n)(x₁ + x₂ ⋯ x_n)) if n is odd.

For $q=3^r$ ($r\in \mathbb{N}$), denote by ${𝔽}_q$ the finite field of order $q$ and for a positive integer $m\ge 2$, let ${𝔽}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive normal element $\alpha$ such that $f(\alpha )$ is a primitive element, where $f(x)=a{x}^2+bx+c$, with $a,b,c\in {𝔽}_{q^m}$ satisfying $b^2\ne ac$ in ${𝔽}_{q^m}$.

The aim of the paper is to introduce a new class of rings — the inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA $A$ a derivative series of IGWAs, $A\to A^{\prime }\to A^{″}\to \cdots \to A^{(\alpha )}\to \cdots$, is attached where $\alpha$ is an arbitrary ordinal. In general, all rings $A^{(\alpha )}$ are distinct. A new construction of rings, the inner $(\sigma ,\tau ,a)$-extension of a ring, is introduced (where $\sigma$ and $\tau$ are endomorphisms of a ring $D$ and $a\in D$).

Let ${𝔽}_{q^n}$ be an extension of the field ${𝔽}_q$ of degree $n,$ where $q=p^k$ for some positive integer $k$ and prime $p.$ In this paper, we establish a sufficient condition for the existence of a primitive element $\alpha \in {𝔽}_{q^n}$ such that ${\alpha }^2+\alpha +1$ is also primitive as well as a primitive normal element $\alpha$ of ${𝔽}_{q^n}$ over ${𝔽}_q$ such that ${\alpha }^2+\alpha +1$ is primitive.