Department of Mathematics

Let a pointed Hopf algebra H, over a field K, be generated as an algebra by the finite group G = G(H) of group-like elements of H and by a coideal A, which satisfies the normalizing condition AK[G] = K[G]A. If char K = 0 we additionally assume that H is generated by group-like and skew primitive elements. It is proved that if A is a semiprime H-module algebra and A acts on A finitely and nilpotently with the semiprime subalgebra of invariants A A , then A satisfies a polynomial identity if and only if A A satisfies a polynomial identity. Applications of this result to actions of concrete Hopf algebras on semiprime algebras are described.

Let R be a semiprime algebra over a field ‫ދ‬ of characteristic zero acted finitely on by a finite-dimensional Lie superalgebra L = L 0 ⊕ L 1 . It is shown that if L is nilpotent, [L 0 , L 1 ] = 0 and the subalgebra of invariants R L is central, then the action of L 0 on R is trivial and R satisfies the standard polynomial identity of degree 2 Examples of actions of nilpotent Lie superalgebras, with central invariants and with [L 0 , L 1 ] = 0, are constructed. 1. Preliminaries. If R is an algebra over a field ‫ދ‬ of characteristic = 2 and σ is a ‫-ދ‬linear automorphism of R such that σ 2 = 1, let D 0 = {δ ∈ End ‫ދ‬ (R) | δ(rs) = δ(r)s + rδ(s) and δσ (r) = σ δ(r) for all r, s ∈ R} and D 1 = {δ ∈ End ‫ދ‬ (R) | δ(rs) = δ(r)s + σ (r)δ(s) and δσ (r) = -σ δ(r) for all r, s ∈ R}. Then D 0 ⊕ D 1 is a Lie superalgebra and the elements of D 0 and D 1 are respectively, derivations and skew derivations of R. The superbracket on D we say that L acts on R if there is a homomorphism of Lie superalgebras ψ : L → D 0 ⊕ D 1 , where ψ(L i ) ⊆ D i , for i = 0, 1. Throughout the paper we will simply assume that L ⊆ D 0 ⊕ D 1 identifying the elements of L 0 and L 1 with their images under ψ. It is well known that the homomorphism ψ induces an associative homomorphism from the universal enveloping algebra U(L) to End ‫ދ‬ (R) and its image is finite dimensional if and only if the derivations and skew derivations from L 0 and L 1 are algebraic. In this case we will say that L acts finitely on R. Letting G be the group {1, σ }, we can form the skew group ring H = U(L) * G and H is now a Hopf algebra acting on R. When L acts on R, we define the subalgebra of invariants R L to be the set {r ∈ R | δ(r) = 0, for all δ ∈ L} . Depending upon the context, the symbol [ , ] may represent either the superbracket on L, or the commutator map [r, s] = rssr, where r, s belong to an associative algebra. Inductively, we let L 1 = L and L n+1 = [L n , L] and we say that L is nilpotent if there exists a positive integer N such that L N = 0. If R (resp. L) is an associative algebra (resp. Lie superalgebra) we will let Z(R) (resp. Z(L)) denote its centre. For an element a ∈ R, and automorphism σ of R, ad a (resp. ∂ a ) stands for the inner derivation (inner σ -derivation) adjoint to a, i.e. ad a (x) = axxa (∂ a (x) = ax -σ (x)a).

Let R be an H-module algebra, where H is a pointed Hopf algebra acting on R finitely of dimension N . Suppose that L H = 0 for every nonzero H-stable left ideal of R. It is proved that if R H satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN provided at least one of the following additional conditions is fulfilled: (1) R is semiprime and where [ √ N ] is the greatest integer in √ N .

Let R be an associative unital ring with the unit group U (R). Let S denote one of the following sets: the set of elements of R, of left ideals of R, of principal left ideals of R, or of ideals of R. Then the group U (R) × U (R) acts on the set S by left and right multiplication. In this note we are going to discuss some properties of rings R with a finite number of orbits under the action of U (R) × U (R) on S.

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series. All rings considered in this paper are assumed to be associative but not necessarily with unit. The standard extension of a ring R to a unital ring with the help of the ring of integers is denoted by R 1 . The sets of idempotent elements in R and nilpotent elements in R are denoted by E(R) and N (R) respectively. J. Lambek in introduced the notion of a symmetric ring understood as a unital ring R in which rst = 0 implies rts = 0 for any r, s, t ∈ R, and proved that an equivalent condition on a unital ring R to be symmetric is that 0 for any positive integer n, any elements r 1 , r 2 , . . . , r n ∈ R and any permutation σ of the set 1, 2, . . . , n . D. D. Anderson and V. Camillo in [1] continued the study of rings whose zero

In the paper, we introduce the notion of a nondistributive ring N as a generalization of the notion of an associative ring with unit, in which the addition needs not be abelian and the distributive law is replaced by n0 = 0n = 0 for every element n of N . For a nondistributive ring N , we introduce the notion of a nondistributive ring of left quotients S -1 N with respect to a multiplicatively closed set S ⊆ N , and determine necessary and sufficient conditions for the existence of S -1 N .

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects.

Let R be an H-module algebra, where H is a pointed Hopf algebra acting on R finitely of dimension N . Suppose that L H = 0 for every nonzero H-stable left ideal of R. It is proved that if R H satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN provided at least one of the following additional conditions is fulfilled: (1) R is semiprime and where [ √ N ] is the greatest integer in √ N .

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series. All rings considered in this paper are assumed to be associative but not necessarily with unit. The standard extension of a ring R to a unital ring with the help of the ring of integers is denoted by R 1 . The sets of idempotent elements in R and nilpotent elements in R are denoted by E(R) and N (R) respectively. J. Lambek in introduced the notion of a symmetric ring understood as a unital ring R in which rst = 0 implies rts = 0 for any r, s, t ∈ R, and proved that an equivalent condition on a unital ring R to be symmetric is that 0 for any positive integer n, any elements r 1 , r 2 , . . . , r n ∈ R and any permutation σ of the set 1, 2, . . . , n . D. D. Anderson and V. Camillo in [1] continued the study of rings whose zero

Let a pointed Hopf algebra H, over a field K, be generated as an algebra by the finite group G = G(H) of group-like elements of H and by a coideal A, which satisfies the normalizing condition AK[G] = K[G]A. If char K = 0 we additionally assume that H is generated by group-like and skew primitive elements. It is proved that if A is a semiprime H-module algebra and A acts on A finitely and nilpotently with the semiprime subalgebra of invariants A A , then A satisfies a polynomial identity if and only if A A satisfies a polynomial identity. Applications of this result to actions of concrete Hopf algebras on semiprime algebras are described.

Let R be a semiprime algebra over a field ‫ދ‬ of characteristic zero acted finitely on by a finite-dimensional Lie superalgebra L = L 0 ⊕ L 1 . It is shown that if L is nilpotent, [L 0 , L 1 ] = 0 and the subalgebra of invariants R L is central, then the action of L 0 on R is trivial and R satisfies the standard polynomial identity of degree 2 Examples of actions of nilpotent Lie superalgebras, with central invariants and with [L 0 , L 1 ] = 0, are constructed. 1. Preliminaries. If R is an algebra over a field ‫ދ‬ of characteristic = 2 and σ is a ‫-ދ‬linear automorphism of R such that σ 2 = 1, let D 0 = {δ ∈ End ‫ދ‬ (R) | δ(rs) = δ(r)s + rδ(s) and δσ (r) = σ δ(r) for all r, s ∈ R} and D 1 = {δ ∈ End ‫ދ‬ (R) | δ(rs) = δ(r)s + σ (r)δ(s) and δσ (r) = -σ δ(r) for all r, s ∈ R}. Then D 0 ⊕ D 1 is a Lie superalgebra and the elements of D 0 and D 1 are respectively, derivations and skew derivations of R. The superbracket on D we say that L acts on R if there is a homomorphism of Lie superalgebras ψ : L → D 0 ⊕ D 1 , where ψ(L i ) ⊆ D i , for i = 0, 1. Throughout the paper we will simply assume that L ⊆ D 0 ⊕ D 1 identifying the elements of L 0 and L 1 with their images under ψ. It is well known that the homomorphism ψ induces an associative homomorphism from the universal enveloping algebra U(L) to End ‫ދ‬ (R) and its image is finite dimensional if and only if the derivations and skew derivations from L 0 and L 1 are algebraic. In this case we will say that L acts finitely on R. Letting G be the group {1, σ }, we can form the skew group ring H = U(L) * G and H is now a Hopf algebra acting on R. When L acts on R, we define the subalgebra of invariants R L to be the set {r ∈ R | δ(r) = 0, for all δ ∈ L} . Depending upon the context, the symbol [ , ] may represent either the superbracket on L, or the commutator map [r, s] = rssr, where r, s belong to an associative algebra. Inductively, we let L 1 = L and L n+1 = [L n , L] and we say that L is nilpotent if there exists a positive integer N such that L N = 0. If R (resp. L) is an associative algebra (resp. Lie superalgebra) we will let Z(R) (resp. Z(L)) denote its centre. For an element a ∈ R, and automorphism σ of R, ad a (resp. ∂ a ) stands for the inner derivation (inner σ -derivation) adjoint to a, i.e. ad a (x) = axxa (∂ a (x) = ax -σ (x)a).

In the paper, we introduce the notion of a nondistributive ring N as a generalization of the notion of an associative ring with unit, in which the addition needs not be abelian and the distributive law is replaced by n0 = 0n = 0 for every element n of N . For a nondistributive ring N , we introduce the notion of a nondistributive ring of left quotients S -1 N with respect to a multiplicatively closed set S ⊆ N , and determine necessary and sufficient conditions for the existence of S -1 N .

In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk for finite dimensional restricted Lie algebras. Our main result is Theorem 13 If R is a graded-reduced ring of characteristic p > 2 acted on by a finitely generated restricted K-Lie superalgebra L, then R L = 0. We can then use Theorem 13 to prove Corollary 15 Let R be a reduced algebra over a field K of characteristic p > 2 acted on by a finite dimensional restricted K-Lie superalgebra L and let H = u(L)#G, where G is the group of order 2 with the natural action on L. If R H satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN , where N is the dimension of H.

In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk for finite dimensional restricted Lie algebras. Our main result is Theorem 13 If R is a graded-reduced ring of characteristic p > 2 acted on by a finitely generated restricted K-Lie superalgebra L, then R L = 0. We can then use Theorem 13 to prove Corollary 15 Let R be a reduced algebra over a field K of characteristic p > 2 acted on by a finite dimensional restricted K-Lie superalgebra L and let H = u(L)#G, where G is the group of order 2 with the natural action on L. If R H satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN , where N is the dimension of H.