We give a new characterization of the bounded nilradical. Using the interplay between this and previous characterizations, we prove that the bounded nilradical is a graded ideal if the ring is graded. This allows us to completely determine the bounded nilradical of skew polynomial and skew Laurent polynomial rings in terms of information in the coefficient ring.

Odd degree Chebyshev polynomials over a ring of modulo 2 have two kinds of period. One is an “orbital period”. Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate affecting a factor of the ring, we can observe an orbit over the ring. The “ orbital period ” is a period of the orbit. The other is a “degree period”. It is observe...

The paper studies algebraic shift equivalence of matrices over n-variable polynomial rings over a principal ideal domain D(n ≤ 2). It is proved that in the case n = 1, every non-nilpotent matrix over D[x] is algebraically strong shift equivalent to a nonsingular matrix. In the case n = 2, an example of non-nilpotent matrix over R[x, y, z] = R[x][y, z], which can not be algebraically shift equiv...

In this thesis we will have a look at algebraic curves in the projective plane over an arbitrary algebraically closed field k. Using the resultant of polynomial rings over k we define intersection multiplicities and prove Bézout’s Theorem for effective divisors. We define singularities and inflexion points and count their number depending on the degree of the curve, using the Hessian of a curve.

We prove the following generalization of the classical ShephardTodd-Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring A := kpij [x1, · · · , xn]. Then the fixed subring A has finite global dimension if and only if G is generated by quasireflections. In this case the fixed subring A is isomorphic a skew polynomial ring with possibly different pi...

0.1. Related results. Krstić-McCool proved that GL3(A) is not finitely presented if there is an epimorphism from A to F [t] for some field F [Krs-Mc]. Suslin proved that SLn(A[t1, . . . , tk]) is generated by elemetary matrices if n ≥ 3, A is a regular ring, and K1(A) ∼= A× [Su]. GrunewaldMennicke-Vaserstein proved that Sp2n(A[t1, . . . , tk]) is generated by elementary matrices if n ≥ 2 and A ...

Modular reduction in extended finite fields and polynomial rings is presented, which once implemented works for any random reduction polynomial without changes of the hardware. It is possible to reduce polynomials of whatever degree. Based on the principal defined, two example RTL architectures are designed, and some useful features are noted furthermore. The first architecture is sequential an...

Two actions of the Hecke algebra of type A on the corresponding polynomial ring are introduced. Both are deformations of the natural action of the symmetric group on polynomials, and keep symmetric functions invariant. An explicit combinatorial formula is given for the resulting graded characters on the coinvariant algebra.