In this paper, we first investigate some properties of an orthogonal complement of a subsemimodule and give some necessary and sufficient conditions for every subsemimodule to be equal to the orthogonal complement of its orthogonal complement. Then we obtain some necessary and sufficient conditions for a set of standard orthogonal vectors to be extended to a standard orthogonal basis of a semimodule. As applications, we present two equivalent descriptions for a semiring to be an exact ring by using terminologies of the orthogonal complements and extending orthogonal subsets of semimodules, respectively.

中文翻译：

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半模的正交补集和扩展正交子集

在本文中，我们首先研究了子半模的正交补的一些性质，并给出了每个子半模等于其正交补的正交补的充分必要条件。然后我们得到了将一组标准正交向量扩展为半模的标准正交基的充分必要条件。作为应用，我们分别使用正交补集的术语和扩展半模的正交子集，提出了两个等价的半环描述为精确环。