A commutative ring is a variety, because its definition consists only of universally quantified identities:(adsbygoogle = window.adsbygoogle || []).push({});

g+(h+k) = (g+h)+k

g+0=g

(-g) + g = 0

g + h = h + g

g(hk) = (gh)k

g(h+k) = gh+gk

1g = g

gh = hg

where (-g) denotes the additive inverse of g.

Adding a new predicate symbol "(1/...)" defined by

(1/g)g = 1

and adding this identity to the list, a new variety is created, whose members are fields along with the trivial ring.

Why is it so important to exclude the trivial ring in the definition of a field (by the requirement [itex]0\neq1[/itex] or [itex]|G|\ge2[/itex]) even though fields are not varieties? Is there any value to the structure defined above (the smallest variety containing all rings)?

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# Varieties, rings, fields

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