Some commutativity criteria for prime rings with differential identities on Jordan ideals

In this article, we present some commutativity theorems for a ring R equipped with a generalized derivation satisfying certain differential identities on Jordan ideals of R. Some related results for prime rings are also discussed. Finally; we provide examples to show that the assumed restrictions are not superfluous.     

Derivation modules of orthogonal duals of hyperplane arrangements

Let A be an n × d matrix having full rank n. An orthogonal dual A^⊥ of A is a (d-n) × d matrix of rank (d−n) such that every row of A^⊥ is orthogonal (under the usual dot product) to every row of A. We define the orthogonal dual for arrangements by identifying an essential (central) arrangement of d hyperplanes in n-dimensional space with the n × d matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. When n ≥ 5, we show that if the matroid (or the lattice of intersection) of an n-dimensional essential arrangement contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement ^⊥ has projective dimension at least ⌈ n(n+2)/4 ⌉ - 3.Hal Schenck partially supported by NSF DMS 03-11142, NSA MDA 904-03-1-0006, and ATP 010366-0103.     

因子von Neumann代数上导子的等价刻画

设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△:R→R在C可导,即△(AB)=△(A)B+A△(B),A,B∈R,AB=C当且仅当存在导子δ:R→R,使得△(A)=δ(A)+△(I)A,A∈R.没有I_1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B(X),使得ran(C)或ker(C)在X中可补,则可加映射△:B(X)→B(X)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.     

可换环上上三角矩阵代数的局部若当导子和局部若当自同构

Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and that every local Jordan automorphism of Tn（R） is a Jordan automorphism. As applications, we show that local derivations and local automorphisms of Tn （R） are inner.     

‘It takes me longer,but I understand better’ – student feedback on structured derivations

In this article, we present the results from a qualitative study on students' initial reactions to the use of structured derivations in mathematics education. Our findings suggest that the approach increases the clarity of solutions and facilitates debugging of proofs. It also has potential to increase students' self-perceived level of understanding. Our findings indicate that the main drawbacks experienced by the students are related to time and length. Nevertheless, the overall feedback on the approach was found to be positive, thus encouraging further use of structured derivations in mathematics education.