A non-zero $\mathbb{F}$-valued $\mathbb{F}$-linear map on a finite dimensional $\mathbb{F}$-algebra is called an $\mathbb{F}$-valued trace if its kernel does not contain any non-zero ideals. However, given an $\mathbb{F}$-algebra such a map may not always exist. We find an infinite class of finite-dimensional commutative $\mathbb{F}$-algebras which admit an $\mathbb{F}$-valued trace. In fact, in these cases, we explicitly construct a trace map. The existence of an $\mathbb{F}$-valued trace on a finite dimensional commutative $\mathbb{F}$-algebra induces a non-degenerate bilinear form on the $\mathbb{F}$-algebra which may be helpful both theoretically and computationally. In this article, we suggest a couple of applications of an $\mathbb{F}$-valued trace map of an $\mathbb{F}$-algebra to algebraic coding theory.

翻译：在有限维$\mathbb{F}$-代数上，若一个非零的$\mathbb{F}$-线性映射的核不包含任何非零理想，则称其为$\mathbb{F}$-值迹。然而，给定一个$\mathbb{F}$-代数，这样的映射可能并不总是存在。我们找到了一类无限族有限维交换$\mathbb{F}$-代数，它们允许$\mathbb{F}$-值迹的存在。事实上，在这些情形中，我们显式构造了一个迹映射。有限维交换$\mathbb{F}$-代数上$\mathbb{F}$-值迹的存在性诱导出该$\mathbb{F}$-代数上一个非退化双线性型，这在理论和计算中可能均有助益。本文我们提出了$\mathbb{F}$-代数的$\mathbb{F}$-值迹映射在代数编码理论中的若干应用。