Every virtually cyclic group Γ that surjects onto the infinite dihedral group D∞ contains an index two subgroup Π of the form H ⋊α Z. We show that the Waldhausen Nil-group of Γ vanishes if and only if the Farrell Nil-group of Π vanishes. 1. Statement of results. The Bass Nil-groups, Farrell Nil-groups, and Waldhausen Nil-groups appear respectively as pieces in the computation of the algebraicK-...

Given a semigroup S, we prove that if the upper nilradical Nil∗(R) is homogeneous whenever R is an S-graded ring, then the semigroup S must be cancelative and torsion-free. In case S is commutative the converse is true. Analogs of these results are established for other radicals and ideals. We also describe a large class of semigroups S with the property that whenever R is a Jacobson radical ri...

We introduce a new notion of regularity for structured ring spectra, and we prove, in the presence of this condition, the existence of new Blumberg–Mandell-type localization sequences for algebraic K-theory. In particular, we confirm a conjecture of Ausoni–Rognes on the behavior of K(KO) as well as a natural generalization of this conjecture to K(Tmf). An early success of Quillen’s algebraic K-...

In continuation of the recent developments on extended reversibilities on rings, we initiate here a study on reversible rings with involutions, or, in short, ∗-reversible rings. These rings are symmetric, reversible, reflexive, and semicommutative. In this note we will study some properties and examples of ∗-reversible rings. It is proved here that the polynomial rings of ∗-reversible rings may...

We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.

This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group. 1. Statement of Results Let R be a ring with unit. For an integer q, let KqR be the algebraic K-group of Bass and Quillen. Bass ...

In this note we find the least order of a noncommutative even square ring and that it is nil having characteristic four. to prove main result given in mainly use suitable examples.

It has recently been shown that a minimal reversible nonsymmetric ring order 256 answering question originally posed in paper on taxonomy of 2-primal rings. In different work, questions rings having to do with this were also answered. work it is abelian reflexive nonsemicommutative 256, related left open these previous works. here [Formula: see text] such ring. This consequence the main result ...

In this paper, we give a new method for constructing LCD codes. We employ group rings and well known map that sends ring elements to subring of the n×n matrices obtain Our construction guarantees our codes are also codes, namely, ideals in ring. show with certain condition on element v, one can construct non-trivial Moreover, by adding more constraints reversible. present many examples binary w...

Japanese morning glory (Ipomoea nil) is a representative plant lacking a yellow-flowered cultivar, although a few wild Ipomoea species contain carotenoids in their petals such as Ipomoea sp. (yellow petals) and I. obscura (pale-yellow petals). In the present study, carotenoid composition and the expression patterns of carotenogenic genes during petal development were compared among I. nil, I. o...