elias001
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- TL;DR Summary
- I would like to know what the word/phrase: "identify" and its variations means in different math context and how is used.
The following are taken from
Lectures on Algebra by Shreeram Shankar Abhyankar
Commutative Algebra Volume 2 By Oscar Zariski, Pierre Samuel
Abstract Algebra An Introduction by Thomas Hungerford
Concepts in Abstract Algebra by Charles Lanski
Background
Questions
The passages quoted above either contain the word "identify", "identified", or "identifying" (highlight in blue). I have taken from various sources at various levels. I would like to know how either of the phrase is used and what it means. I am not sure if it means in the sense of two things being equal or does it mean in the sense of isomorphism?
Thank you in advance
Lectures on Algebra by Shreeram Shankar Abhyankar
Commutative Algebra Volume 2 By Oscar Zariski, Pierre Samuel
Abstract Algebra An Introduction by Thomas Hungerford
Concepts in Abstract Algebra by Charles Lanski
Background
Let ##R## be an integral domain and let ##S## be this set of pairs:
$$S=\{(a,b)\mid a,b\in R, b\neq 0\}$$
Define a relation ##\sim## on the set ##S## by
$$(a,b)\sim (c,d)\text{ means }ad-bc \text{ in }R$$.
Theorem 10.30 Let ##R## be an integral domain. Then there exists a field ##F## whose elements are of the form ##a/b## with ##a,b\in R## and ##b\neq 0_R##, subject to the equality condition
$$\frac{a}{b}= \frac{c}{d}\text{ in }F\text{ if and only if }ad=bc\text{ in }R$$
Addition and multiplication in ##R## are given by
$$\frac{a}{b} +\frac{c}{d} =\frac{ad+bc}{bd}, \frac{a}{b}\cdot\frac{c}{d} =\frac{ac}{bd},$$
The set of elements in ##F## of the form ##a/1_R(a\in R)## is an integral domain isomorphic to ##R##.
....The ring ##R## will be ##\color{blue}{identified}## with its isomorphic copy in ##F##. Then we can say that ##R## is the subset of ##F## consisting of elements of the form ##a/1_R##. The field ##F## is called the field of quotients of ##R##.
Exercise: If $##m\in \mathbb{Z}##, find a maximal multiplicatively closed set ##T\subset \mathbb{Z}## that satisfies ##T\cap\{m^k\mid k\geq 0\}=\emptyset## and ##\color{blue}{identify}## ##\mathbb{Z}_T## in ##\mathbb{Q}## if ##m=p^n## is a prime power.
...This makes ##R_S## into a ring and ##u\mapsto u/1## gives the "canonical" ring homomorphism
$$\phi: R\to R_S.$$
Clearly
$$\phi(S)\subset U(R_S)\text{ and } \mathrm{ker}(\phi)=[0:S]_R$$
where we recall that for any ideal ##I## in ##R## we have defined
$$[I:S]_R=\{r\in R:rs\in I\text{ for some }s\in S\}$$
$$=\text{the isolated }S-\text{component of }I\text{ in }R.$$
The localization of ##R## at ##S_R(R)## is called the total quotient ring of ##R## and denoted by ##\text{QR}(R)##. Clearly ##[0:S_R(R)]_R=0## and hence the canonical map of ##R## into ##\text{QR}(R)## is injective. By ##\color{blue}{"identifying"}## every ##u\in R## with ##u/1\in \text{QR}(R)##, we may and we shall regard ##\text{QR}(R)## to be an overring of ##R##.
Since the ideal ##\cap_{n=0}^{\infty}\mathfrak{m}^n=(0)##, which is the closure of ##0## in ##\hat{A}##, it follows that the quotient ring ##A_S## may be ##\color{blue}{identified}## with a subring of the completion ##\hat{A}##.
Questions
The passages quoted above either contain the word "identify", "identified", or "identifying" (highlight in blue). I have taken from various sources at various levels. I would like to know how either of the phrase is used and what it means. I am not sure if it means in the sense of two things being equal or does it mean in the sense of isomorphism?
Thank you in advance