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：若非零$\mathbb{F}$-值$\mathbb{F}$-线性映射在有限维$\mathbb{F}$-代数上的核不含任何非零理想，则称其为$\mathbb{F}$-值迹。然而，对于给定的$\mathbb{F}$-代数，此类映射未必总存在。我们找到了一类无穷族有限维交换$\mathbb{F}$-代数，其允许$\mathbb{F}$-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，会在该$\mathbb{F}$-代数上诱导一个非退化双线性型，这在理论与计算层面均可能有所助益。本文提出了$\mathbb{F}$-代数上$\mathbb{F}$-值迹映射在代数编码理论中的若干应用。