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Semirings as defined on Wolfram MathWorld don’t have either an additive or multiplicative identity; they are semigroup objects in the category of commutative semigroups.
The same is true of Kazimierz Glazek’s A guide to the literature on semirings and their applications in mathematics and information sciences
Thank you, it seems very logical to me now. I will use the term rig for the structure with the two identities from now on.
I’ve seen four different definitions of a “semiring” out there, depending on the author:
The problem is already there in the definition of a ring, as some authors define a ring to be a semigroup object in the category of abelian groups, while other authors define a ring to be a monoid object in the category of commutative monoids
*define a ring to be a monoid object in the category of abelian groups
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