Explore the vast range of titles available.

We generalize a classical result about derivation pairs on function algebras. Specifically, we describe the forms of derivation pairs on rings and rngs (non-unital rings) which are not assumed to be commutative. The proofs are based on knowledge of the solutions of the sine addition formula on a semigroup. Examples are given to illustrate the results.

Let S and B be two unital ∗ -algebras such that S has a nontrivial projection. In the present article, we demonstrate, under certain restrictions that if a bijective map Δ : S → B satisfies Δ ( M ⋄ N ∘ W ) = Δ ( M ) ⋄ Δ ( N ) ∘ Δ ( W ) for all M , N , W ∈ S , then Δ is a ∗ -preserving ring isomorphism. As an application, we will describe these mappings on factor von Neumann algebras.

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of the local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra A , we define a group homomorphism Hdet : Aut d g ( A ) → k × , called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra A G to be Gorenstein, where A is a homologically smooth DG algebra such that H ( A ) is a Noetherian AS-Gorenstein graded algebra and G is a finite subgroup of Aut d g ( A ) . Especially, we can apply this result to DG down-up algebras and non-trivial DG free algebras generated in two degree-one elements.

The Heisenberg double D q ( E 2 ) of the quantum Euclidean group O q ( E 2 ) is the smash product of O q ( E 2 ) with its Hopf dual U q ( e 2 ) . For the algebra D q ( E 2 ), explicit descriptions of its prime, primitive and maximal spectra are obtained. All the prime factors of D q ( E 2 ) are presented as generalized Weyl algebras. As a result, we obtain that the algebra D q ( E 2 ) has no finite-dimensional representations, and D q ( E 2 ) cannot have a Hopf algebra structure. The automorphism groups of the quantum Euclidean group and its Heisenberg double are determined. Some centralizers are explicitly described via generators and defining relations. This enables us to give a classification of simple weight modules and the so-called a -weight modules over the algebra D q ( E 2 ).

Let R be a Noetherian integral domain which is also an algebra over ℚ (ℚ is the field of rational numbers). Let σ be an endo-morphism of R and δ a σ-derivation of R. We recall that a ring R is a weak (σ, δ)-rigid ring if a(σ(a)+ δ(a)) N(R) if and only if a N(R) for a R (N(R) is the set of nilpotent elements of R). With this we prove that if R is a Noetherian integral domain which is also an algebra over ℚ, σ an automorphism of R and δ a σ-derivation of R such that R is a weak (σ, δ)-rigid ring, then N(R) is completely semiprime.

The article explores research findings akin to Amitsur’s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In most results related to rings and semirings, Birkenmeier’s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.

In the present paper, we prove that a prime ring with center satisfies , the standard identity in four variables if admits a non-identity automorphism σ such that ( ]) for all in some non-central Lie ideal of whenever either char( )> or char( )=0, where is a fixed positive integer.

We study the automorphism group of the algebra $\\co_q(M_n)$ of n × n generic quantum matrices. We provide evidence for our conjecture that this group is generated by the transposition and the subgroup of those automorphisms acting on the canonical generators of $\\co_q(M_n)$ by multiplication by scalars. Moreover, we prove this conjecture in the case when n = 3.

McCoy proved that for a right ideal A of S = R[x1, . . ., xk] over a ring R, if rS(A) ≠ 0 then rR(A) ≠ 0. We extend the result to the Ore extensions, the skew monoid rings and the skew power series rings over non-commutative rings and so on.

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism σ of a prime one-sided noetherian ring R is injective whenever the image σ(R) contains an essential left ideal L of R. If, in addition, σ(L)=L, then σ is an automorphism of R. Examples showing that the assumptions imposed on R cannot be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.