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:
- 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 $(R \setminus \{0\},*,1)$.
- 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!
$\endgroup$2 Answers
$\begingroup$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 $(R\setminus\{0\},*,1)$.
$\endgroup$ $\begingroup$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 $X$.
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