# Are Naturals a semiring, Integers a ring, and Rationals a field?

This seems a really simple way to remember which is which, but if it was right, I'd expect an equally base name for the one for Naturals than semiring ("rig" doesn't count). I think this is how it goes for operators:

semiring: $+ *$
e.g. Naturals, no inverses

ring: $+ - *$
e.g. Integers, semiring with additive inverse (enabling subtraction)

field: $+ - * /$
e.g. Rationals, ring with multiplicative inverse (enabling division)

Are there any wrinkles I've missed that mean this isn't quite right?

• That's right. ${}$ – anon May 19 '18 at 3:59
• – Morgan Rodgers May 19 '18 at 4:10
• There are of course the ramificatoins whether or not "ring" means that multiplication is abelean and/or has a unit element. – Hagen von Eitzen May 19 '18 at 4:25
• @HagenvonEitzen But having abelian multiplication and/or a unit certainly doesn't disqualify a structure from being a ring – Morgan Rodgers May 19 '18 at 4:37