Polynomial rings over finite dimensional rings

Triangularization over finite-dimensional division rings using the reduced trace

In this paper we study triangularization of collections of matrices whose entries come from a finite-dimensional division ring. First, we give a generalization of Guralnick's theorem to the case of finite-dimensional division rings and then we show that in this case the reduced trace function is a suitable alternative for trace function by presenting two triangularization results. The first one...

Polynomial Rings over Pseudovaluation Rings

Let R be a ring. Let σ be an automorphism of R. We define a σ-divided ring and prove the following. (1) Let R be a commutative pseudovaluation ring such that x ∈ P for any P ∈ Spec(R[x,σ]) . Then R[x,σ] is also a pseudovaluation ring. (2) Let R be a σ-divided ring such that x ∈ P for any P ∈ Spec(R[x,σ]). Then R[x,σ] is also a σ-divided ring. Let now R be a commutative Noetherian Q-algebra (Q i...

Codes over finite quotients of polynomial rings

Codes over finite quotient of polynomial rings

Differential Polynomial Rings of Triangular Matrix Rings

Polynomial Equivalence of Finite Rings

We prove that Zpn and Zp[t]/(t) are polynomially equivalent if and only if n ≤ 2 or p = 8. For the proof, employing Bernoulli numbers, we provide the polynomials which compute the carry-on part for the addition and multiplication in base p. As a corollary, we characterize finite rings of p elements up to polynomial equivalence.