Miguel Ferrero

Higher Derivations and a Theorem By Herstein

In this paper we extend to the higher derivations a well-known result proved by Herstein concerning derivations in prime rings. We prove results which imply that every Jordan higher derivation of a 2-torsion-free semiprime ring is a higher derivation.

Unitary strongly prime rings and related radicals

A unitary strongly prime ring is defined as a prime ring whose central closure is simple with identity element. The class of unitary strongly prime rings is a special class of rings and the corresponding radical is called the unitary strongly prime radical. In this paper we prove some results on unitary strongly prime rings. The results are applied to study the unitary strongly prime radical of a polynomial ring and also R-disjoint maximal ideals of polynomial rings over R in a finite number of indeterminates. From this we get relations between the Brown–McCoy radical and the unitary strongly prime radical of polynomial rings. In particular, the Brown–McCoy radical of R[X] is equal to the unitary strongly prime radical of R[X] and also equal to S(R)[X], where S(R) denotes the unitary strongly prime radical of R, when X is an infinite set of either commuting or non-commuting indeterminates. For a PI ring R this holds for any set X.

Partial Skew Polynomial Rings: Prime and Maximal Ideals

In this article, we consider rings R with a partial action α of a cyclic infinite group G on R. We define partial skew polynomial rings as natural subrings of the partial skew group ring R ⋆α G. We study prime and maximal ideals of a partial skew polynomial ring when the given partial action α has an enveloping action.

On the ideal structure of right distributive rings

In t roduc t ion A right distributive ring, right D-ring for short, is a ring whose lattice of right .ideals is distributive. It is well-known that the class of commutative D-domains coincides with the class of Priifer domains. Noncommutative right D-rings were investigated in a paper of Stephenson [13]. Brungs (51 proved that right D-domains are locally right chain rings. So the class of right chain rings is an interesting class of examples (see [3] and the literature quoted therein). Recently two papers by Mazureli and Puczyiowski respectively Mazurek showed that some features for right chain rings can be carried over to right D-rings WI, 191). The purpose of this paper is t o study the structure of right ideals in right D-rings. In

HIGHER DERIVATIONS OF SEMIPRIME RINGS

1 we first recall some results from [9]. We introduce the relevant condition (MP). A right D-ring is said to satisfy (MP) if it has a completely prime ideal contained in the Jacobson radical. Some examples of right D-rings satisfying (MP) are given. In 52 we first introduce some notions such as right multiplicative ideals, waists and the right associated completely prime ideal P,(I) of a right multi-

Partial Skew Polynomial Rings and Goldie Rings

ABSTRACT In this paper we study higher derivations of prime and semiprime rings satisfying linear relations. We extend several results known for algebraic derivations, and we prove some other results.

Actions of Inverse Semigroups on Algebras

We prove that if R is a semiprime ring and α is a partial action of an infinite cyclic group on R, then R is right Goldie if and only if R[x; α] is right Goldie if and only if R⟨x; α⟩ is right Goldie, where R[x; α] (R⟨x; α⟩) denotes the partial skew (Laurent) polynomial ring over R. In addition, R⟨x; α⟩ is semiprime while R[x; α] is not necessarily semiprime.

A note on locally nilpotent derivations

In this article we prove that if S is a faithfully projective R-algebra and H is a finite inverse semigroup acting on S as R-linear maps such that the fixed subring S H = R, then any partial isomorphism between ideals of S which are generated by central idempotents can be obtained as restriction of an R-automorphism of S and there exists a finite subgroup of automorphisms G of S with S G = R.

CLOSED AND PRIME IDEALS IN PARTIAL SKEW GROUP RINGS OF ABELIAN GROUPS

Let R be a commutative reduced, Z-torsion free ring. Let d and δ be two locally nilpotent derivations of R which commute, a an element of R. We prove that the derivation ad + δ is locally nilpotent if and only if d(a) = 0.

Prime and maximal ideals in polynomial rings

In this paper we study partial actions of abelian groups on a ring R having an enveloping action. Among other results we prove that if α is such a partial action and (T, β) its globalization, then there exists a one-to-one correspondence between closed (respectively, R-disjoint prime) ideals of R ⋆α G and closed (respectively, T-disjoint prime) ideals of T ⋆β G. We also prove that there exists a one-to-one correspondence between closed (respectively, R-disjoint prime) ideals of R ⋆α G and closed (respectively, -disjoint prime) ideals of , where is the left α-quotient ring of R. Finally, we use these results to study strongly prime ideals and nonsingular prime ideals of R ⋆α G.