Undergrad Meaning of "identify" & variations in different math context

[elias001]

TL;DR

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

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

 

[fresh_42]

You identify all the time when you deal with quotients. We identify $\dfrac{1}{2}$ with e.g. $\dfrac{2}{4}$ or $\dfrac{-3}{-6}$ because $1\cdot 4=2\cdot 2$ and $1\cdot (-6)=2\cdot (-3),$ or $2\cdot (-6)=4\cdot (-3).$ That's it.

And if we consider cosets, say in the ring $\mathbb{R}[x]/\bigl\langle x^2+1 \bigr\rangle$ then we identify the coset $x+\bigl\langle x^2+1 \bigr\rangle$ with $i$ since we identified the coset $0+\bigl\langle x^2+1 \bigr\rangle$ with $0$ itself.

 

[elias001]

@fresh_42 "identify" is not the same as "being equivalent" in mathematics usage?

 

[PeroK]

@fresh_42 "identify" is not the same as "being equivalent" in mathematics usage?

Words and phrases like "identify" and "being equivalent" have no universal meaning in mathematics. Their meaning depends on the context. For example, "equivalent" is often used to describe two statements that are true if and only if one implies the other. Or, you could have two equivalent definitions - of continuity, for example.

I would think in terms of the specific mathematics. These words and phrases are often used to give you some insight over and above the raw mathematical statements. There will always be something totally precise behind the explanations.

 

[elias001]

@PeroK I have a different post about the phrase "passage to the quotient", and when I see these phrases being used, I feel like either I am failing in language comprehension or there is some math associated with these phrases that I don't understand. I would never be able to confidently used either of the phrases "identify" or "passage to the quotient" because i don't know their precise meaning in mathematics and how and in what circumstances they can be used.

 

[fresh_42]

@fresh_42 "identify" is not the same as "being equivalent" in mathematics usage?

As @PeroK has already mentioned, it depends.

Myself, I like to consider the rational numbers as equivalence classes and use exactly the definition of localizations in rings:
$$
\dfrac{a}{b}\sim \dfrac{c}{d} \Longleftrightarrow a\cdot d= b\cdot c.
$$
My favorite example is that two quarters of a pie are not the same as half a pie, since they have one more cut.

But if you start a thread here in which you claim this, you will earn a shitstorm insisting that one half is exactly two quarters and this was an equation and not an equivalence. I tried. If you want to be picky, then you could say that $\dfrac{a}{b}=\dfrac{c}{d}$ in $\mathbb{Q}$ and $\dfrac{a}{b}\sim\dfrac{c}{d}$ in $\left(\mathbb{Z}\times \mathbb{Z}/\sim\;\right) =\left(\mathbb{Z}\setminus \{0\}\right)^{-1}\,\mathbb{Z}.$

Or take the example of $1=0.\overline{9}.$ Do we identify them? Yes, of course. But are they the same or equivalent representations of the same number? Also, yes to both.