Onofrio Mario Di Vincenzo

On the graded identities ofM1,1(E)

We determine theSn×Sm-cocharacterXn,m of the algebraM1,1(E) and prove that theT2-ideal of its graded identities is generated by the polynomialsy1y2−y2y1 andz1z2z3+z3z2z1.

Graded Polynomial Identities for Tensor Products by the Grassmann Algebra

Abstract Let 𝕂 be a field of characteristic zero, and R be a G-graded 𝕂-algebra. We consider the algebra R ⊗ E, then deduce its G × ℤ2-graded polynomial identities starting from the G-graded polynomial identities of R. As a consequence, we describe a basis for the ℤ n × ℤ2-graded identities of the algebras M n (E). Moreover we give the graded cocharacter sequence of M 2(E), and show that M 2(E) is PI-equivalent to M 1,1(E) ⊗ E. This fact is a particular case of a more general result obtained by Kemer.

Posner's second theorem, multilinear polynomials and vanishing derivations

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d and δ non-zero derivations of R , f (x 1 ,…, x n ) a multilinear polynomial over K .If then f ( x 1 ,…, x n is central-valued on R.

GRADED POLYNOMIAL IDENTITIES OF VERBALLY PRIME ALGEBRAS

Let F be a field and let E be the Grassmann algebra of an infinite dimensional F-vector space. For any p,q ∈ ℕ, the algebra Mp,q(E) can be turned into a ℤp+q × ℤ2-algebra by combining an elementary ℤp+q-grading with the natural ℤ2-grading on E. The tensor product Mp,q(E) ⊗ Mr,s(E) can be turned into a ℤ(p+q)(r+s) × ℤ2-algebra in a similar way. In this paper, we assume that F has characteristic zero and describe a system of generators for the graded polynomial identities of the algebras Mp,q(E) and Mp,q(E) ⊗ Mr,s(E) with respect to these new gradings. We show that this tensor product is graded PI-equivalent to Mpr+qs,ps+qr(E). This provides a new proof of the well known Kemers PI-equivalence between these algebras. Then we classify all the graded algebras Mp,q(E) having no non-trivial monomial identities, and finally calculate how many non-isomorphic gradings of this new type are available for Mp,q(E).

Quadratic Central Differential Identities on a Multilinear Polynomial

Let K be a commutative ring with unity, R a prime K-algebra, Z(R) the center of R, d and δ nonzero derivations of R, and f(x 1,…, x n ) a multilinear polynomial over K. If [d(f(r 1,…, r n )), δ (f(r 1,…, r n ))] ∈ Z(R), for all r 1,…, r n ∈ R, then either f(x 1,…, x n ) is central valued on R or {d, δ} are linearly dependent over C, the extended centroid of R, except when char(R) = 2 and dim C RC = 4.

Derivations on multilinear polynomials in semiprime rings

Let R, he a 2-torsion free semiprime K-algebra with unity, d a non-zero derivation of Rand a non-zero multilinear polynomial over K. Suppose that, for every is zero or invertible in R. Then either R. is a division ring, or is a central polynomial for R, or , for every , the left Utumi quotient ring of R, that is there exists a central idempotent e of Usuch that d vanishes identically on eUand is central in (1-e)U. Moreover the last conclusion holds if and only if , for every ri .

On the *-Polynomial Identities of a Class of *-Minimal Algebras

Let F be an infinite field. We consider certain block-triangular algebras with involution U n , with n ∈ ℕ, having minimal *-exponent. We describe their *-polynomial identities, and in case char.F = 0, their structure as a T *-ideal under the action of general linear groups. These goals are achieved by means of Y-proper polynomials. We also compute explicitly the irreducible modules occurring in the decomposition of B Y (U 3) and their multiplicities.

ALGEBRAS SATISFYING THE POLYNOMIAL IDENTITY [x1,x2][x3,x4,x5]=0

Let be a field of characteristic zero, and the variety of associative unitary algebras defined by the polynomial identity [x1,x2][x3,x4,x5]=0. This variety is one of the several minimal varieties of exponent 3 (and all proper subvarieties are of exponents 1 and 2). We describe asymptotically its proper subvarieties. More precisely, we define certain algebras ℛ2k for any k∈ℕ and show that if is a proper subvariety of , then the T-ideal of its polynomial identities is asymptotically equivalent to the T-ideal of the identities of one of the algebras , E, ℛ2k or ℛ2k⊕E, for a suitable k∈ℕ. We give also another description relating the T-ideals of the proper subvarieties of with the polynomial identities of upper triangular matrices of a suitable size.

On the Existence of the Graded Exponent for Finite Dimensional Z p -graded Algebras

Let F be an algebraically closed field of characteristic zero, and let A be an associative unitary F-algebra graded by a group of prime order. We prove that if A is finite dimensional then the graded exponent of A exists and is an integer. Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, 85100 Potenza, Italia e-mail: onofrio.divincenzo@unibas.it Dipartimento di Matematica, Università degli Studi di Bari, via Orabona 4, 70125 Bari, Italia e-mail: nardozza@dm.uniba.it Received by the editors February 12, 2009; revised June 7, 2009. Published electronically May 30, 2011. Partially supported by MUR and Università di Bari. AMS subject classification: 16R50, 16R10, 16W50.

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.