Müller generalized in [12] the notion of a Frobenius extension to left (right) quasi-Frobenius extension and proved the endomorphism ring theorem for these extensions. Recently, Guo observed in [9] that for a ring homomorphism φ : R → S, the restriction of scalars functor has to induction functor S ⊗R − : RM → SM as right ”quasi” adjoint if and only if φ is a left quasi-Frobenius extension. In ...

In this paper we show that finite rings for which the extension theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and López-Permouth.

Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover resul...

We consider codes defined over an affine algebra A = R[X1, . . . , Xr]/ 〈t1(X1), . . . , tr(Xr)〉, where ti(Xi) is a monic univariate polynomial over a finite commutative chain ring R. Namely, we study the A−submodules of A (l ∈ N). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. Some codes over Frobenius local ...

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I. Definitions of the operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Definitions of N and F for curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. The monodromy ope...

Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring, and the conditio...

Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover resul...

Overview. My work is in the area of commutative algebra, and is motivated by the connections between algebra and geometry. Commutative algebra is the study of commutative rings (e.g. polynomial rings over fields and their quotients) and modules over these rings, while algebraic geometry is the study of finitely many polynomial equations in finitely many unknowns. The set of all solutions to suc...

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of code...