Wagner Cortes

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.

Partial Skew Polynomial Rings and Goldie Rings

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.

Actions of Inverse Semigroups on Algebras

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.

Skew Polynomial Extensions over Zip Rings

In this article, we study the relationship between left (right) zip property of 𝑅 and skew polynomial extension over 𝑅, using the skew versions of Armendariz rings.

Partial Skew Polynomial Rings and Jacobson Rings

In this article we consider rings R with a partial action α of an infinite cyclic group G on R. We generalize the well-known results about Jacobson rings and strongly Jacobson rings in skew polynomial rings and skew Laurent polynomial rings to partial skew polynomial rings and partial skew Laurent polynomial rings.

Partial Actions on Categories

In this article, we introduce the definition of partial action of groups on R-semicategories, generalizing the similar notion of partial actions on categories. Our main result is Theorem 4.6 on the existence of an enveloping action for a partial action of a group on a small R-semi-category. We also define the notion of partial skew category, furthermore proving results similar to the ones in [7].

Partial Skew Polynomial Rings Over Semisimple Artinian Rings

Let R be a semisimple Artinian ring with a partial action α of ℤ on R, and let R[x; α] be the partial skew polynomial ring. By the classification of the set E of all minimal central idempotents in R into three different types, a complete description of the prime radical of R[x; α] is given. Moreover, it is shown that any nonzero prime ideal of R[x; α] is maximal and is either principal or idempotent. In the case where α is of finite type, it is shown that R[x; α] is a semiprime hereditary ring.