We consider the periodicity of recursive sequences defined by linear homogeneous recurrence relations of arbitrary order, when they are reduced modulo a positive integer m. We show that the period of such a sequence with characteristic polynomial f can be expressed in terms of the order of ω = x + f as a unit in the quotient ring ‫ޚ‬ m [ω] = ‫ޚ‬ m [x]/ f. When m = p is prime, this order can be ...

Let m,n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : R −→ R be a skew derivation of R and E(x) = D(xm+n+r)−D(xm)xn+r − xmD(xn)xr − xm+nD(xr). We prove that if E(x) = 0 for all x ∈ L, then D is a usual derivation of R or R satisfies s4(x1, . . . , x4), the standard identity of degree 4.

In this paper we consider the K-theory of smooth algebraic stacks, establish λ and Adams operations and show that the higher K-theory of such stacks is always a pre-λ-ring and is a λ-ring if every coherent sheaf is the quotient of a vector bundle. As a consequence we are able to define Adams operations and absolute cohomology for smooth algebraic stacks satisfying this hypothesis. We also defin...

Throughout, R will be a commutative ring with identity with total quotient ring T (R), group of units U(R), set of zero-divisors Z(R), and Jacobson radical J(R). For a; b 2 R, we de...ne three associate relations: 1. We say a and b are associate, denoted a » b, if ajb and bja, (a) = (b). 2. We say a and b are strongly associate, denoted a 1⁄4 b, if there exists a u 2 U (R) such that a = ub. 3. ...

In [2], Billera proved that the R-algebra of continuous piecewise polynomial functions (C0 splines) on a d-dimensional simplicial complex 1 embedded in Rd is a quotient of the Stanley–Reisner ring A1 of 1. We derive a criterion to determine which elements of the Stanley–Reisner ring correspond to splines of higher-order smoothness. In [5], Lau and Stiller point out that the dimension of C k (1)...

Let n = (n1, . . . , nr). The quotient space Pn := S n1× · · ·×Snr/(x ∼ −x) is what we call a projective product space. We determine the integral cohomology ring H∗(Pn) and the action of the Steenrod algebra on H∗(Pn;Z2). We give a splitting of ΣPn in terms of stunted real projective spaces, and determine the ring K∗(Pn). We relate the immersion dimension and span of Pn to the much-studied sect...

Let G be a finite group and let k be a field. Our purpose is to investigate the simple modules for the double Burnside ring kB(G,G). It turns out that they are evaluations at G of simple biset functors. For a fixed finite group H, we introduce a suitable bilinear form on kB(G,H) and we prove that the quotient of kB(−, H) by the radical of the bilinear form is a semi-simple functor. This allows ...

Our task here is to recall part of the theory of orders and ideals in quaternion algebras. Some of the theory makes sense in the context of B/K a quaternion algebra over a field K which is the quotient field of a Dedekind ring R. For our purposes K will always be a number field, or the completion of a number field at a finite prime, and R will be the ring of integers of K. (Nevertheless, we sha...

Different choices of basis yield (non-canonically) isomorphic Cox rings and any of them is a “total” coordinate ring for X in the sense that, (1) The homogeneous coordinate rings ⊕∞ s=0H (X, sD) of all images of X via complete linear systems φD : X → P(H(X,D)) are subalgebras of Cox(X). (2) If the Cox ring is a finitely generated k-algebra then X can be obtained as a quotient of an open set of ...

Consider the projective coordinate ring of the GIT quotient (P)//SL(2), with the usual linearization, where n is even. In 1894, Kempe proved that this ring is generated in degree one. In [HMSV2] we showed that, over Q, the relations between degree one invariants are generated by a class of quadratic relations — the simplest binomial relations — with the exception of n = 6, where there is a sing...