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arXiv · 1212.5358

Exact rings and semirings

David Wilding·Marianne Johnson·Mark Kambites
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Abstract

We introduce and study an abstract class of semirings, which we call exact semirings, defined by a Hahn-Banach-type separation property on modules. Our motivation comes from the tropical semiring, and in particular a desire to understand the often surprising extent to which it behaves like a field. The definition of exactness abstracts an elementary property of fields and the tropical semiring, which we believe is fundamental to explaining this similarity. The class of exact semirings turns out to include many other important examples of both rings (proper quotients of principal ideal domains, matrix rings and finite group rings over these and over fields), and semirings (the Boolean semiring, generalisations of the tropical semiring, matrix semirings and group semirings over these).

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David Wilding, Marianne Johnson, Mark Kambites. 2013-05-08. Exact rings and semirings. https://doi.org/10.1016/j.jalgebra.2013.05.005

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