# A more succinct definition of subfield

I'm reading textbook Algebra by Saunders MacLane and Garrett Birkhoff in which a subfield is defined as

A subset of a field $$F$$ is a subfield if and only if it is closed under the operations multiplicative unit, subtraction, multiplication, and multiplicative inverse (of non-zero elements).

My questions:

1. From this definition of subring, i.e.

A subring of a ring $$(\mathrm{R},+, *, 0,1)$$ is a subset $$\mathrm{S}$$ of $$\mathrm{R}$$ that preserves the structure of the ring, i.e. a ring $$(\mathrm{S},+, *, 0,1)$$ with $$\mathrm{S} \subseteq \mathrm{R}$$. Equivalently, it is both a subgroup of $$(\mathrm{R},+, 0)$$ and a submonoid of $$(\mathrm{R}, *, 1)$$.

I understand "Equivalently, it is both a subgroup of $$(\mathrm{R},+, 0)$$ and a submonoid of $$(\mathrm{R}, *, 1)$$" as

A subset $$S$$ is a subring of $$R$$ if and only if $$S$$ is an additive subgroup of $$(R,+,0)$$ and $$S \setminus \{0\}$$ is a multiplicative submonoid of $$(R \setminus \{0\},*,1)$$.

1. Inspired by above definition. I've come up with a more succinct definition of subfield, i.e.

A subset $$E$$ of a field $$(F,+, *, 0,1)$$ is a subfield if and only if $$E$$ is an additive subgroup of $$(F,+,0)$$ and $$E \setminus \{0\}$$ is a multiplicative subgroup of $$(F \setminus \{0\},*,1)$$.

Could you please verify if my understanding is correct? Thank you so much for your help!

A little correction: Your second formulation (version of subfield definition) is correct, but the first one about subrings is not true in general: $$(R\setminus\{0\},*,1)$$ itself need not be a monoid (i.e. closed under multiplication), as the ring $$R$$ can have zero divisors or $$R\setminus\{0\}$$ might be empty.
Saying $$(R\setminus\{0\},*,1)$$ is a monoid (i.e. a submonoid of $$(R,*,1)$$) already implies $$1\neq 0$$ and $$R$$ has no zero divisors. In this case (only), $$(S,*,1)$$ is a submonoid of $$(R,*,1)$$ iff $$(S\setminus\{0\},*,1)$$ is a submonoid of $$(R\setminus\{0\},*,1)$$.
Yes both are correct. You probably notice the pattern in all these definitions: a sub-floop of a floop $$X$$ is a subset $$Y$$ of $$X$$ that is still a floop with the operations it inherits from $$X$$.