# Algebraic K-theory with commutative semirings?

My question is basically given in the title: Are there any references for a generalization of algebraic K-theory to the scenario where the domain of the functors consists of commutative semirings (provided this has been done at all)?

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I dont know of any references, but at least one can say the obvious generalisation of higher K theory isnt as nice. The forgetful functor U Semiring to Set still has a left adjoint F and thus there is a cotriple FU. So for a semiring there is still a simplicial semiring, and one can apply the GL functor and take simplicial homotopy groups. The problem though is that GL is much less interesting for semirings. –  Jason Polak Aug 15 '12 at 22:37
A trivial way to do it: define the $K$-theory of a semiring as the $K$-theory of the ring obtained by 'adding' additive inverses. I guess you may have some idea behing your question. Perhapes you should say someting about it in order to possible get interesting answers. –  Fernando Muro Aug 16 '12 at 0:02
People have certainly been interested in the categorified version of your question (jtopol.oxfordjournals.org/content/4/3/…). Do you have some more context for your question? Is there a particular category of modules or something that you want to understand? –  K.J. Moi Aug 16 '12 at 6:07

There are a few papers out there dealing with a slightly different focus - algebraic K-theory over the "field with one element" $\mathbb{F}_1$. Some of the frameworks for $\mathbb{F}_1$ algebra include semirings, so these papers might contain material that covers what you are interested in as well.