Generalized Polynomial Identities Research Articles

Linear identities and commuting maps in rings with involution

. In this paper we extend Martindale's result by showing that the symmetric elements or the skew elements of a prime ring with involution of characteristic not 2 do not satisfy certain linear generalized polynomial identities. As applications, we determine the centralizing additive maps and the commuting traces of biadditive maps on the symmetric elements of a prime ring.

Prime rings with PI rings of constants

It is shown that if the ring of constants of a restricted differential lie algebra with a quasi-Frobenius inner part satisfies a polynomial identity (PI) then the original prime ring has a generalized polynomial identity (GPI). If additionally the ring of constants is semiprime then the original ring is PI. The case of a non-quasi-Frobenius inner part is also considered.

Orders in primitive rings with non-zero socle and posner's theorem

By introducing a modification of the notion of classical ring of quotients we extend, on the one hand, Goldie’s theorem to non-singular prime rings with uniform one­sided ideals and, on the other hand, Posner's theorem to prime rings with a generalized polynomial identity.

Azumaya Algebras with Involution, Polarizations, and Linear Generalized Identities

An algebra R with anti-isomorphism (*) is shown to be Azumaya if (*) is given by an element of R ⊗ Rop; in particular, this is the case if the canonical map R ⊗ CRop → EndC(R) is onto. Consequently, the existence of a strict polarization often implies that an algebra is Azumaya. On the other hand, all simple rings have polarizations, and algebras with involution of the second kind have polarizations. These results are obtained via the theory of generalized polynomial identities.

Left annihilators characterized by GPIs

Let R R be a semiprime ring with extended centroid C C , U U the right Utumi quotient ring of R R , S S a subring of U U containing R R and ρ 1 {\rho _1} , ρ 2 {\rho _2} two right ideals of R R . In the paper we show that l S ( ρ 1 ) = l S ( ρ 2 ) {l_S}({\rho _1}) = {l_S}({\rho _2}) if and only if ρ 1 {\rho _1} and ρ 2 {\rho _2} satisfy the same generalized polynomial identities (GPIs) with coefficients in S C SC , where l S ( ρ i ) {l_S}({\rho _i}) denotes the left annihilator of ρ i {\rho _i} in S S . As a consequence of the result, if ρ \rho is a right ideal of R R such that l R ( ρ ) = 0 {l_R}(\rho ) = 0 , then ρ \rho and U U satisfy the same GPIs with coefficients in the two-sided Utumi quotient ring of R R .

Generalized polynomial identities and rings which are sums of two subrings

Generalized polynomial identities and rings which are sums of two subrings

A short proof of Martindale's theorem on GPIs

We give a short proof of the following theorem of Martindale: If a prime ring R satisfies a nontrivial generalized polynomial identity, then the ring RC, where C is the extended centroid of R, contains a minimal idempotent e such that eRCe is finite dimensional over C.

Differential identities with automorphisms and antiautomorphisms, I

Let R be a prime associative ring with the extended centroid C. Assume that R satisfies a nontrivial differential identity with automorphisms and antiautomorphisms. It is shown here that R must satisfy a nontrivial ordinary generalized polynomial identity (without derivations, automorphisms, and antiautomorphisms). When this is combined with Martindale's result on generalized polynomial identities, it follows that the central closure RC of R is a primitive ring with nonzero socle and its skew field is finite dimensional over C.

Extended centroids of power series rings

Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study.

Minimal ∗-differential identities in prime rings

It is known that a prime ring which satisfies a polynomial identity with derivations applied to the variables must satisfy a generalized polynomial identity, but not necessarily a polynomial identity. In this paper we determine the minimal identity with derivations which can be satisfied by a non-PI prime ringR. The main result shows, essentially, that this identity is the standard identityS3 withD applied to each variable, whereD = ad(y) fory inR, y2 = 0, andy of rank one in the central closure ofR.

On subnormal subgroups of skew fields

If D is a skew field with infinite center Z and [ D: Z]= ∞ then any subnormal subgroup of D∗ which satisfies a Laurent generalized polynomial identity is central.

Minimal differential identities in prime rings

It is known that a prime ring which satisfies a polynomial identity with derivations applied to the variables must satisfy a generalized polynomial identity, but not necessarily a polynomial identity. In this paper we determine the minimal identity with derivations which can be satisfied by a non-PI prime ringR. The main result shows, essentially, that this identity is the standard identityS3 withD applied to each variable, whereD = ad(y) fory inR, y2 = 0, andy of rank one in the central closure ofR.

Differential identities in prime rings with involution

Let R R be a prime ring with involution. Using work of V. K. Kharchenko it is shown that any generalized identity for R R involving derivations of R R and the involution of R R is a consequence of the generalized identities with involution which R R satisfies. In obtaining this result, a generalization, to rings satisfying a GPI, of the classical theorem characterizing inner derivations of finite-dimensional simple algebras is required. Consequences of the main theorem are that in characteristic zero no outer derivation of R R can act algebraically on the set of symmetric elements of R R , and if the images of the set of symmetric elements under the derivations of R R satisfy a polynomial relation, then R R must satisfy a generalized polynomial identity.

*-Generalized polynomial identities of finite dimensional central simple algebras

LetD be a finite dimensional central simple algebra with involution * of the first kind. We prove (in a fixed number of variables) the ideal of *-GPI’s ofD is generated by a finite collection of elements of the form [v ij ,v pq ] andv ij −v ij * where thev ij ’s are first degree generalized polynomials.

Entartete Steinerkegelschnitte in nichtpapposschen Desarguesebenen

In a projective plane the set of all points which are on homologous lines of two projective, non-perspective, different pencils with self-homologous common line is called a degenerated Steiner (d. S.) conic. An analytic description for d. S. conics is given. Amitsur's theorem about generalized polynomial identities in division rings finds two geometrical applications: (1) in the discussion of the normal form problem for d. S. conics and (2) in answering the question if a d. S. conic allows beside the defining generation further generations. We study the intersection of d. S. conics with lines and notice that a d. S. conic is never identical with a set of absolute points of a polarity.

Monomial conditions on prime rings

The study of pivotal monomials (and related conditions) is continued and extended, with the aim of studying carefully a situation generalizing Martindale's theory of prime rings with generalized polynomial identity. This is used to describe various classes of rings in terms of simple elementary sentences. The focus is on prime “Johnson” rings, which play a crucial role in our characterizations. It turns out that these rings can be characterized in terms of generalized pivotal monomials, thereby yielding a theory similar to that of [17].

Generalized polynomial identities, III

Generalized polynomial identities, III

General polynomial identities. II

A generalized polynomial is a formal polynomial with coefficients in a noncommutative ring (so that the coefficients are interspersed with the indeterminates); a generalized monomial of a generalized polynomial is a sum of all monomials in which the indeterminates occur in the same order. A generalized polynomial identity (GI) of R is R-proper if one of its generalized monomials is not a GI of R. The two main results of this paper are: (i) Any GI of a primitive ring P is P soc P-improper, where soc P is the socle of P; (ii) if R satisfies a GI that is proper for every nonzero homomorphic image of R, then R is a PI-algebra (in the usual sense). These theorems have several immediate applications: 1. (i) If some element r in R is a root of an algebraic equation f( λ) = 0 with f′( r) invertible, f′ the formal derivative of f, and if the centralizer of r is a PI-algebra, then R is a PI-algebra (most of this was already known by M. Smith); 2. (ii) If R has two left ideals A and B that are PI-algebras, then A + B is a PI-algebra. The main results of the paper are also given in the context of rings with involution.

Structure of Rings with Involution Applied to Generalized Polynomial Identities

In [14, §4], some theorems were obtained about generalized polynomial identities in rings with involution, but the statements had to be weakened somewhat because a structure theory of rings with involution had not yet been developed sufficiently to permit proofs to utilize enough properties of rings with involution. In this paper, such a theory is developed. The key concept is that of the central closure of a ring with involution, given in § 1, shown to have properties analogous to the central closure of a ring without involution. In § 2, the theory of primitive rings with involution, first set forth by Baxter-Martindale [5], is pushed forward, to enable a setting of generalized identities in rings with involution which can parallel the non-involutory situation.

Generalized polynomial identities

Generalized polynomial identities