Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.     

On Skew Triangular Matrix Rings

For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring T _n (R, α). By using an ideal theory of a skew triangular matrix ring T _n (R, α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x; α]/ ? x ⁿ  ?, for each positive integer n, where R[x; α] is the skew polynomial ring, and ? x ⁿ  ? is the ideal generated by x ⁿ .     

On Finite Saturated Chains of Overrings

If a domain R, with quotient field K, has a finite saturated chain of overrings from R to K, then the integral closure of R is a Prüfer domain. An integrally closed domain R with quotient field K has a finite saturated chain of overrings from R to K with length m ≥ 1 iff R is a Prüfer domain and |Spec(R)| =m + 1. In particular, we prove that a domain R has a finite saturated chain of overrings from R to K with length dim(R) iff R is a valuation domain and that an integrally closed domain R has a finite saturated chain of overrings from R to K with length dim (R) +1 iff R is a Prüfer domain with exactly two maximal ideals such that at most one of them fails to contain every non-maximal prime. The relationship with maximal non-valuation subrings is also established.     

Recovering Quivers from Derived Quiver Representations

We compute Balmer's prime spectrum for the derived category of quiver representations for a finite ordered quiver with the vertex-wise tensor product and show that it does not recover the quiver. We then associate an algebra to every k-linear triangulated tensor category and show that the path algebra can be recovered in this way.     

Prime Submodules and a Sheaf on the Prime Spectra of Modules

We define and investigate a sheaf of modules on the prime spectra of modules, and it is shown that there is an isomorphism between the sections of this sheaf and the ideal transform module.     

Right Centralizers of Semiprime Rings

Let R be a semiprime ring with Q _ml (R) the maximal left ring of quotients of R. Suppose that T: R → Q _ml (R) is an additive map satisfying T(x ²) = xT(x) for all x ∈ R. Then T is a right centralizer; that is, there exists a ∈ Q _ml (R) such that T(x) = xa for all x ∈ R.     

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L _R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L _R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L _R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L _R (E) coincide.     

数论中切比雪夫不等式的一点补充

利用Wallis不等式,得到素数计算函数的一个下界估计.