Minimal degree of standard identities of matrix algebras with symplectic graded involution

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.

Полудифференцирования в первичных кольцах

Let $\mathscr{R}$ be a prime ring with the extended centroid $\mathscr{C}$ and the Matrindale quotient ring $\mathscr{Q}$. An additive mapping $\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}$ is called a semiderivation associated with a mapping $\mathscr{G}: \mathscr{R}\rightarrow \mathscr{R}$, whenever $ \mathscr{F}(xy)=\mathscr{F}(x)\mathscr{G}(y)+x\mathscr{F}(y)= \mathscr{F}(x)y+ \mathscr{G}(x)\mathscr{F}(y) $ and $ \mathscr{F}(\mathscr{G}(x))= \mathscr{G}(\mathscr{F}(x))$ holds for all $x, y \in \mathscr{R}$. In this manuscript, we investigate and describe the structure of a prime ring $\mathscr{R}$ which satisfies $\mathscr{F}(x^m\circ y^n)\in \mathscr{Z(R)}$ for all $x, y \in \mathscr{R}$, where $m,n \in \mathbb{Z}^+$ and $\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}$ is a semiderivation with an~automorphism $\xi$ of $\mathscr{R}$. Further, as an application of our ring theoretic results, we discussed the nature of $\mathscr{C}^*$-algebras. To be more specific, we obtain for any primitive $\mathscr{C}^*$-algebra $\mathscr{A}$. If an anti-automorphism $ \zeta: \mathscr{A} \to \mathscr{A}$ satisfies the relation $(x^n)^\zeta+x^{n*}\in \mathscr{Z}(\mathscr{A})$ for every ${x,y}\in \mathscr{A},$ then $\mathscr{A}$ is $\mathscr{C}^{*}-\mathscr{W}_{4}$-algebra, i.\,e., $\mathscr{A}$ satisfies the standard identity $\mathscr{W}_4(a_1,a_2,a_3,a_4)=0$ for all $a_1,a_2,a_3,a_4\in \mathscr{A}$.

A Note on Differential Identities in Prime and Semiprime Rings

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

On Certain Identities with Automorphisms on Lie Ideals in Prime and Semiprime Rings

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]t – [x, y]m [α([x, y]),[x, y]]n [x, y]s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s4, the standard identity of degree 4. We also extend the results to semiprime rings.

A note on automorphisms of lie ideals in prime rings

In the present paper, we prove that a prime ring R with center Z satisfies s4, the standard identity in four variables if R admits a non-identity automorphism σ such that (uσ,u]vσ+vσ[uσ,u])n∈Z for all u,v in some non-central Lie ideal L of R whenever either char(R)>n or char(R)=0, where n is a fixed positive integer.

Generalized Derivations on Lie Ideals and Power Values on Prime Rings

Let R be a non-commutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, L a non-central Lie ideal of R, G a non-zero generalized derivation of R.If [G(u), u](1) R satisfies the standard identity s(2) there exists γ ∈ C such that G(x) = γx for all x ∈ R.