Fuzzy semirings

In this paper we initiate the study of fuzzy semirings and fuzzy A-semimodules where A is a semiring and A-semimodules are representations of A. In particular, semirings all of whose ideals are idempotent, called fully idempotent semirings, are investigated in a fuzzy context. It is proved, among other results, that a semiring A is fully idempotent if and only if the lattice of fuzzy ideals of A is distributive under the sum and product of fuzzy ideals. It is also shown that the set of proper fuzzy prime ideals of a fully idempotent semiring A admits the structure of a topological space, called the fuzzy prime spectrum of A.     

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.     

Quotient Rings of Ore Extensions with More than One Indeterminate

Let R be a domain and R[X; D] the Ore extension of R by a sequence D of derivations of R. If D has length ≥ 2, we show that the symmetric Utumi quotient ring of R[X; D] is U _s (R)[X; D], where U _s (R) is the symmetric Utumi quotient ring of R. Consequently, X-inner automorphisms of R[X; D] are induced by units of U _s (R) and the extended centroid of R[X; D] consists of those elements α in the center of U _s (R) such that δ(α) = 0 for all δ ? D. These extend the known results for free algebras.     

A Koszul complex over skew polynomial rings

We construct a Koszul complex in the category of left skew polynomial rings associated with a flat endomorphism that provides a finite free resolution of an ideal generated by a Koszul regular sequence.     

On skew polynomial rings over locally nilpotent rings

We will show that skew polynomial rings in several variables over locally nilpotent rings cannot contain nonzero idempotent elements. We will also prove that such rings are Brown–McCoy radical.     

Projective ideals of skew polynomial rings over HNP rings

Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S?=?R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.     

幂等元半环簇P

幂等元半环簇和幂等元分配半环簇依次记为I,ID.满足附加恒等式xyx x xyx=xyx的幂等元半环簇的子簇记为P.本文主要刻划了P中成员的一些性质,并对P∩ID中的部分成员进行了次直积分解.     

A Note on Compositions of Derivations of Prime Rings

Abstract

In this note we discuss when compositions of derivations can be nonzero derivations of prime rings. In particular we answer two questions posed by Lanski.     

半环的星结构

引入了半环的星元素和星理想等概念,并且讨论了半环的星结构,得到了半环的星结构定理.     

On the structure of semi-invariant polynomials in Ore extensions

Let D be an X-outer S-derivation of a prime ring R, where S is an automorphism of R. The following is proved among other things: The degree of the minimal semi-invariant polynomial of the Ore extension R[X;S,D] is ν if charR=0, and is p^kν for some k0 if charR=p2, where ν is the least integer ν1 such that S^νDS^−ν−D is X-inner. A similar result holds for cv-polynomials. These are done by introducing the new notion of k-basic polynomials for each integer k0, which enable us to analyze semi-invariant polynomials inductively.     

Derivations of Tensor Product Algebras

It is known that under certain finite dimensionality condition the derivation algebra of tensor product of two algebras can be obtained in terms of the derivation algebras and the centroids of the involved algebras. We extend this theorem to infinite dimensional case and as an application, we determine the derivation algebra of the fixed point algebra of the tensor product of two algebras, with respect to the tensor product of two finite order automorphisms. These provide the framework for calculating the derivations of some infinite dimensional Lie algebras.     

Characterization of Lie Derivations on Prime Rings

Let ? be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent P. It is shown that, under some mild conditions, an additive map L on ? satisfies L([A, B]) = [L(A), B] + [A, L(B)] whenever AB = 0 (resp., AB = P) if and only if it has the form L(A) = ?(A) + h(A) for all A ∈ ?, where ? is an additive derivation on ? and h is an additive map into its center.     

On Solutions for Derivations of a Noetherian k-algebra and Local Simplicity

We introduce a general notion of solution for a Noetherian differential k-algebra and study its relationship with simplicity, where k is an algebraically closed field; then we analyze conditions under which such solutions may exist and be unique, with special emphasis in the cases of k-algebras of finite type and formal series rings over k. Using that notion we generalize a criterion for simplicity due to Brumatti–Lequain–Levcovitz and give a geometric characterization of that; as an application we give a new proof of a classification theorem for local simplicity due to Hart and obtain a general result for simplicity of formal series rings over k.