# Some remarks on non projective Frobenius algebras and linear codes

## TL;DR

This paper extends the concept of Frobenius algebras by removing the projectivity condition, enabling the construction of new finite Frobenius rings and exploring their implications for linear codes and MacWilliams identities.

## Contribution

It introduces a modified definition of Frobenius algebras over Frobenius rings, broadening the class of rings and codes that can be studied.

## Key findings

- Frobenius algebras over Frobenius rings are themselves Frobenius rings.
- Frobenius finite rings are characterized as Frobenius finite algebras over their characteristic subrings.
- Discussion of generalized MacWilliams identities in this new context.

## Abstract

With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over their characteristic subrings. From the perspective of linear codes, our work expands one's options to construct new finite Frobenius rings from old ones. We close with a discussion of generalized versions of the MacWilliams identities that may be obtained in this context.