Luisa Carini

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x1, …, xn) a noncentral multilinear polynomial over C. If δ(G(f(r1, …, rn))f(r1, …, rn)) = 0 for all r1, …, rn ∈ R, then f(x1, …, xn)2 is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.

Formulas for the expansion of the plethysms s 2[ s (a,b) ] and s 2[ s (n k ) ]

Abstract Building on the ideas of Carbonara, Remmel, and Yang who, recently, gave explicit formulas for the Schur function expansion of the plethysms s 2 [ S μ ] and s 1 2 [ S μ ] where μ is a hook shape, we derive explicit formulas for the Schur function expansion of the plethysms s 2 [ S μ ] and s 1 2 [ S μ ] where μ has two rows or two columns. These formulas generalize classical formulas of Littlewood for the Schur function expansions of s 2 [ S ( n ) ] and s 1 2 [ S ( n ) ]. We also derive explicit formulas for s 2 [ S λ ] and s 1 2 [ S λ ] where λ = ( n k ) for some k and n .

Derivations on Lie ideals in semiprime rings

We prove that ifR is a semiprime 2-torsion free ring with a derivationd andU a Lie ideal ofR such thatd2 (U)=0, thend(U)⊂Z(R), the center ofR.

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x 1,…, x n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r 1,…, r n ∈ R, a[F 2(f(r 1,…, r n )), f(r 1,…, r n )] = 0, then one of the following statements hold: 1. a = 0; 2. There exists λ ∈C such that F(x) = λx, for all x ∈ R; 3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c 2 ∈ C; 4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c 2 ∈ C.

Identities with Product of Generalized Derivations of Prime Rings

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, f(x1,…,xn) a multilinear polynomial over C which is not an identity for R, F and G two non-zero generalized derivations of R. If F(u)G(u)=0 for all u ∈ f(R)= {f(r1,…,rn): ri∈ R}, then one of the following holds: (i) There exist a, c ∈ U such that ac=0 and F(x)=xa, G(x)=cx for all x ∈ R; (ii) f(x1,…,xn)2 is central valued on R and there exist a, c ∈ U such that ac=0 and F(x)=ax, G(x)=xc for all x ∈ R; (iii) f(x1,…,xn) is central valued on R and there exist a,b,c,q ∈ U such that (a+b)(c+q)=0 and F(x)=ax+xb, G(x)=cx+xq for all x ∈ R.

The hilbert series of the polynomial identities for the tensor square of the grassmann algebra

LetE be the Grassmann (or exterior) algebra of an infinite-dimensional vector space over a fieldK of characteristic 0. We compute the Hilbert series of the relatively free algebraFm(varE⊗KE)=K(xl,...,xm) /(T(E⊗KE)∩K〈xl,...,xm〉) of the variety of algebrasvarE⊗KE, whereT(E⊗KE) is the set of all polynomial identities forE⊗KEE from the free associative algebraK〈x1,x2…〉.

The cocharacter sequence of the finite dimensional grassmann algebra

Let Vk be a k-dimensional vector space and let E(Vk ) be the finite dimensional Grass-mann algebra over a field F. We compute the sequence of cocharacters of the nilpotent algebra E∗ (Vk ) = E(Vk ) − F.

Identities with Product of Generalized Skew Derivations on Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, and C its extended centroid. Suppose that F, G are generalized skew derivations of R, with the same associated automorphism, and f(x1,…, xn) a noncentral multilinear polynomial over C with n noncommuting variables, such that for all r1,…, rn ∈ R. Then we describe all possible forms of F and G.

On Some Generalized Identities with Derivations on Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, Z(R) the center of R, f(x1,…,xn) a non-central multilinear polynomial over K, d and δ derivations of R, a and b fixed elements of R. Denote by f(R) the set of all evaluations of the polynomial f(x1,…,xn) in R. If a[d(u),u] + [δ (u),u]b = 0 for any u ∈ f(R), we prove that one of the following holds: (i) d = δ = 0; (ii) d = 0 and b = 0; (iii) δ = 0 and a = 0; (iv) a, b ∈ Z(R) and ad + bδ = 0. We also examine some consequences of this result related to generalized derivations and we prove that if d is a derivation of R and g a generalized derivation of R such that g([d(u),u]) = 0 for any u ∈ f(R), then either g = 0 or d = 0.

Plethysmsfor representations of lie superalgebras with applications to P. I algebras

We introduce plethysms for representations of general linear Lie superalgebras and of hook symmetric functions and show that the known decompositions of ordinary symmetric functions into a sum of Schur functions can be used for obtaining similar decompositions in the supercase. We apply these results to algebras with polynomial identities. The main result on P.I. algebras is the computation of the double Hilbert (or Poincare) series of the polynomial identities for the tensor square E⊗E of the Grassmann algebra E of an infinite-dimensional vector space.