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Thank you, it’s very usefull. The word rng is used for semigroup objects in the category of abelian groups. Thus we could use:
semi-ring for a monoid object in the category of commutative semigroups
semi-rng for semigroup objects in the category of commutative semigroups
rg for semigroup objects in the category of commutative monoids
rig for monoid objects in the category of commutative monoids
It seems to make sense once you have understood the logic. However, it doesn’t solve the problems for rings.
There is a similar problem with the definition of an algebra. Given a commutative (unital) ring $R$, are $R$-algebras magma objects in $R$-modules, semigroup objects in $R$-modules, or monoid objects in $R$-modules? The multiplication in Jordan algebras, Lie algebras, and Leibniz algebras are not associative, and neither are they unital. But free algebras on a set of generators are by definition associative and unital.
This somewhat carries down to rings, since Lie rings are defined to be Lie $\mathbb{Z}$-algebras, even though Lie rings are not associative or unital, and rings are by definition associative and unital.
As pointed out in another thread (here) the following sentence does not make much sense:
The nLab uses the second definition to define a semiring, and the fourth definition to define a rig.
(Firstly there is no such global policy nor could it practically be enforced even if one were to wish for it, and secondly it seems circular in itself that the entry titled “semiring” would effectively refer to itself this way.)
So I have replaced this sentence now by what the subsequent sentence suggests was the intent here:
If one adopts the second definition to define a semiring, and the fourth definition to define a rig, then…
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