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Of course, commutative algebra is a fundamental topic in algebraic geometry, number theory, representation theory, and so on.

However, there are some instances (most obviously tropical geometry) where one wishes to consider commutative rigs (or semirings) instead of commutative rings. Although most of the basic concepts of commutative algebra generalize to this setting, it's not as obvious which of commutative algebra's most famous theorems also generalize. And unfortunately there aren't canonical texts along the lines of Atiyah-Macdonald or Eisenbud or Matsumura.

Does anyone know of a good reference for learning commutative algebra over semirings?

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One starting point would be the paper On Ideals in semirings of K. Iseki and follow on the articles that cite it.

On the other hand "A guide to the literature on semirings and their applications in mathematics and information sciences" by K. Glazek is a guide to the extensive literature on semirings.

Among books cited there:

"Semirings and their applications", "Semirings and affine equations over them: theory and applications", "Power algebras over semirings" by J.S. Golan and "Semirings: algebraic theory and applications in computer science" by U. Hebisch, H.J. Weinert

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