Mathematics

Let R be a ring with a derivation \delta. In this paper, we prove that an analogue of Amitsur's property holds for left T-nilpotent radideals of pseudo-differential operator rings R((x^{-1}; \delta)), where R is a delta-compatible ring. As a direct consequence of this fact, we obtain an alternative characterization of the prime radical of R((x^{-1}; \delta)).

The purpose of this paper is to study infinitesimal H-pseudobialgebra, which is an associative analogy of Lie H-pseudobialgebra. We first define the infinitesimal H-pseudobialgebra and investigate some properties of this new algebraic structure. Then we consider the coboundary infinitesimal H-pseudobialgebra, which is the subclass of infinitesimal H-pseudobialgebra and we obtain the associative Yang-Baxter equation over an associative H-pseudoalgebra. Finally, we found the connection between the (coboundary) infinitesimal H-pseudobialgebra and the (coboundary) Lie H-pseudobialgebra. Meanwhile, the relationship between the associative Yang-Baxter equation and the classical Yang-Baxter equation (over an H-pseudoalgebra) is established.

Let R\to A be a homomorphism of associative rings, and let (\mathcal F,\mathcal C) be a hereditary complete cotorsion pair in R\mathsf{-Mod} . Let (\mathcal F_A,\mathcal C_A) be the cotorsion pair in A\mathsf{-Mod} in which \mathcal F_A is the class of all left A -modules whose underlying R -modules belong to \mathcal F . Assuming that the \mathcal F -resolution dimension of every left R -module is finite and the class \mathcal F is preserved by the coinduction functor \operatorname{Hom}_R(A,-) , we show that \mathcal C_A is the class of all direct summands of left A -modules finitely (co)filtered by A -modules coinduced from R -modules from \mathcal C . Assuming that the class \mathcal F is closed under countable products and preserved by the functor \operatorname{Hom}_R(A,-) , we prove that \mathcal C_A is the class of all direct summands of left A -modules cofiltered by A -modules coinduced from R -modules from \mathcal C , with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from \mathcal F have finite \mathcal F -resolution dimension bounded by k , involves cofiltrations indexed by the ordinal \omega+k . The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra arXiv:0708.3398. In addition, we discuss the n -cotilting and n -tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz-Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.

Rehren proved that a primitive 2-generated axial algebra of Monster type (α,β) has dimension at most eight if α∉{2β,4β} . In this note we construct an infinite-dimensional 2-generated primitive axial algebra of Monster type (2, 1 2 ) over an arbitrary field F with char(F)≠2,3 . This shows that the second special case, α=4β , is a true exception to Rehren's bound.

We establish a bialgebra theory for anti-flexible algebras in this paper. We introduce the notion of an anti-flexible bialgebra which is equivalent to a Manin triple of anti-flexible algebras. The study of a special case of anti-flexible bialgebras leads to the introduction of anti-flexible Yang-Baxter equation in an anti-flexible algebra which is an analogue of the classical Yang-Baxter equation in a Lie algebra or the associative Yang-Baxter equation in an associative algebra. It is a unexpected consequence that both the anti-flexible Yang-Baxter equation and the associative Yang-Baxter equation have the same form. A skew-symmetric solution of anti-flexible Yang-Baxter equation gives an anti-flexible bialgebra. Finally the notions of an O -operator of an anti-flexible algebra and a pre-anti-flexible algebra are introduced to construct skew-symmetric solutions of anti-flexible Yang-Baxter equation.

Given a block triangular matrix M over a noncommutative ring with invertible diagonal blocks, this work gives two new representations of its inverse M −1 . Each block element of M −1 is explicitly expressed via a quasideterminant of a submatrix of M with the block Hessenberg type. Accordingly another representation for each inverse block is attained, which is in terms of recurrence relationship with multiple terms among blocks of M −1 . The latter result allows us to perform an off-diagonal rectangular perturbation analysis for the inverse calculation of M . An example is given to illustrate the effectiveness of our results.

Let Mold n,d be the moduli of rank d subalgebras of M n over Z . For x∈ Mold n,d , let A(x)⊆ M n (k(x)) be the subalgebra of M n corresponding to x , where k(x) is the residue field of x . In this article, we apply Hochschild cohomology to Mold n,d . The dimension of the tangent space T Mold n,d /Z,x of Mold n,d over Z at x can be calculated by the Hochschild cohomology H 1 (A(x), M n (k(x))/A(x)) . We show that H 2 (A(x), M n (k(x))/A(x))=0 is a sufficient condition for the canonical morphism Mold n,d →Z being smooth at x . We also calculate H i (A, M n (k)/A) for several R -subalgebras A of M n (R) over a commutative ring R . In particular, we summarize the results on H i (A, M n (k)/A) for all k -subalgebras A of M n (k) over an algebraically closed field k in the case n=2,3 .

In this paper, we introduce a new technique in the study of the ∗ -regular closure of some specific group algebras KG inside U(G) , the ∗ -algebra of unbounded operators affiliated to the group von Neumann algebra N(G) . The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form C K (X) ⋊ T Z , where X is a totally disconnected compact metrizable space, T is a homeomorphism of X , and C K (X) stands for the algebra of locally constant functions on X with values on an arbitrary field K . The connection between this class of algebras and a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of ℓ 2 -Betti numbers arising from the lamplighter group, most of them transcendental.

We provide a novel recursive method, which does not require any assumption, to compute the entries of the kth power of a semicirculant matrix. As an application, a method for computing the entries of the kth power of r-circulant matrices is also presented.

Idempotent elements are a well-studied part of ring theory, with several identities of the idempotents in Z/mZ already known. Although the idempotents are not closed under addition, there are still interesting additive identities that can be derived and used. In this paper, we give several new identities on idempotents in Z/mZ . We relate finite sublattices over Z/kZ for all integers k to an infinite lattice that is embedded in the divisibility lattice on N and to each other as sublattices of this infinite lattice. Using this relation, we generalize several identities on idempotents in Z/mZ to those involving idempotents related to these finite sublattices. Finally, as an application of the above idempotent identities, we derive an algorithm for calculating modular exponentiation over Z/mZ .