If we say “classes of non-zero integers modulo $n$â€, why does this not include the $0$ class?
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I suppose this is a bit of a wording question more than anything else - I'm working through group theory and was learning that "the (classes of) non-zero integers modulo $p$ form an Abelian group under multiplication." It's the wording of this that gets me a bit confused. Let's say $p = 3$. From my understanding the group described is meant to contain $[1],[2]$. I completely get why this would be an Abelian group under multiplication. However, if we're looking at all the non-zero integers modulo $3$, it seems to me like the "non-zero" attribute binds to the integer part only. So it's $mathbbZ - 0$ (which would include $3,-3,6,-6,...$) modulo $3$. Hence, since integer multiples of 3 are included in this set, $[0]$ would also be included. But clearly my interpretation can't be the case since including $[0]$ would mean it's not a multiplicative group.
In summary, it's like I'm having trouble with the order of operations here:
Interpretation 1: (set of non-zero integers) modulo $p$
Interpretation 2: set of non-zero (integers modulo $p$)
So am I supposed to interpret it like interpretation 2? Since it seems like interpretation 1 would include $[0]$.
group-theory elementary-number-theory modular-arithmetic
asked Sep 7 at 5:50
rb612
1,754822
add a comment |Â
up vote
2
down vote
favorite
I suppose this is a bit of a wording question more than anything else - I'm working through group theory and was learning that "the (classes of) non-zero integers modulo $p$ form an Abelian group under multiplication." It's the wording of this that gets me a bit confused. Let's say $p = 3$. From my understanding the group described is meant to contain $[1],[2]$. I completely get why this would be an Abelian group under multiplication. However, if we're looking at all the non-zero integers modulo $3$, it seems to me like the "non-zero" attribute binds to the integer part only. So it's $mathbbZ - 0$ (which would include $3,-3,6,-6,...$) modulo $3$. Hence, since integer multiples of 3 are included in this set, $[0]$ would also be included. But clearly my interpretation can't be the case since including $[0]$ would mean it's not a multiplicative group.
In summary, it's like I'm having trouble with the order of operations here:
Interpretation 1: (set of non-zero integers) modulo $p$
Interpretation 2: set of non-zero (integers modulo $p$)
So am I supposed to interpret it like interpretation 2? Since it seems like interpretation 1 would include $[0]$.
group-theory elementary-number-theory modular-arithmetic
asked Sep 7 at 5:50
rb612
1,754822
1
Yes, this is just a bit of sloppy wording. The intent is clearly "the set of equivalence classes modulo $p$ that do not contain zero."
– user7530
Sep 7 at 5:57
If you are going to argue language and syntax then you must consider what role exactly do "modulo n" serve in the phrase ""the (classes of) non-zero integers modulo n" It can't be an adjective because in is after the noun. It could refer to the classes modulo n but then how exactly was the "non-zero integers" cobbled in where it doesn't fit. To my mind an "[integer modulo n]" is a phrase we are suppose to recognize as a modulo class and so a "non-zero [integer modulo n]" is clearly a non-zero modulo class; of which there are two. Admittedly this makes the "class of integers" redundant.
– fleablood
Sep 7 at 6:09
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I suppose this is a bit of a wording question more than anything else - I'm working through group theory and was learning that "the (classes of) non-zero integers modulo $p$ form an Abelian group under multiplication." It's the wording of this that gets me a bit confused. Let's say $p = 3$. From my understanding the group described is meant to contain $[1],[2]$. I completely get why this would be an Abelian group under multiplication. However, if we're looking at all the non-zero integers modulo $3$, it seems to me like the "non-zero" attribute binds to the integer part only. So it's $mathbbZ - 0$ (which would include $3,-3,6,-6,...$) modulo $3$. Hence, since integer multiples of 3 are included in this set, $[0]$ would also be included. But clearly my interpretation can't be the case since including $[0]$ would mean it's not a multiplicative group.
In summary, it's like I'm having trouble with the order of operations here:
Interpretation 1: (set of non-zero integers) modulo $p$
Interpretation 2: set of non-zero (integers modulo $p$)
So am I supposed to interpret it like interpretation 2? Since it seems like interpretation 1 would include $[0]$.
group-theory elementary-number-theory modular-arithmetic
asked Sep 7 at 5:50
rb612
1,754822
I suppose this is a bit of a wording question more than anything else - I'm working through group theory and was learning that "the (classes of) non-zero integers modulo $p$ form an Abelian group under multiplication." It's the wording of this that gets me a bit confused. Let's say $p = 3$. From my understanding the group described is meant to contain $[1],[2]$. I completely get why this would be an Abelian group under multiplication. However, if we're looking at all the non-zero integers modulo $3$, it seems to me like the "non-zero" attribute binds to the integer part only. So it's $mathbbZ - 0$ (which would include $3,-3,6,-6,...$) modulo $3$. Hence, since integer multiples of 3 are included in this set, $[0]$ would also be included. But clearly my interpretation can't be the case since including $[0]$ would mean it's not a multiplicative group.
In summary, it's like I'm having trouble with the order of operations here:
Interpretation 1: (set of non-zero integers) modulo $p$
Interpretation 2: set of non-zero (integers modulo $p$)
So am I supposed to interpret it like interpretation 2? Since it seems like interpretation 1 would include $[0]$.
group-theory elementary-number-theory modular-arithmetic
group-theory elementary-number-theory modular-arithmetic
asked Sep 7 at 5:50
rb612
1,754822
asked Sep 7 at 5:50
rb612
1,754822
asked Sep 7 at 5:50
rb612
1,754822
asked Sep 7 at 5:50
rb612
1,754822
asked Sep 7 at 5:50
rb612
1,754822
1,754822
1
Yes, this is just a bit of sloppy wording. The intent is clearly "the set of equivalence classes modulo $p$ that do not contain zero."
– user7530
Sep 7 at 5:57
If you are going to argue language and syntax then you must consider what role exactly do "modulo n" serve in the phrase ""the (classes of) non-zero integers modulo n" It can't be an adjective because in is after the noun. It could refer to the classes modulo n but then how exactly was the "non-zero integers" cobbled in where it doesn't fit. To my mind an "[integer modulo n]" is a phrase we are suppose to recognize as a modulo class and so a "non-zero [integer modulo n]" is clearly a non-zero modulo class; of which there are two. Admittedly this makes the "class of integers" redundant.
– fleablood
Sep 7 at 6:09
add a comment |Â
1
Yes, this is just a bit of sloppy wording. The intent is clearly "the set of equivalence classes modulo $p$ that do not contain zero."
– user7530
Sep 7 at 5:57
If you are going to argue language and syntax then you must consider what role exactly do "modulo n" serve in the phrase ""the (classes of) non-zero integers modulo n" It can't be an adjective because in is after the noun. It could refer to the classes modulo n but then how exactly was the "non-zero integers" cobbled in where it doesn't fit. To my mind an "[integer modulo n]" is a phrase we are suppose to recognize as a modulo class and so a "non-zero [integer modulo n]" is clearly a non-zero modulo class; of which there are two. Admittedly this makes the "class of integers" redundant.
– fleablood
Sep 7 at 6:09
1
1
Yes, this is just a bit of sloppy wording. The intent is clearly "the set of equivalence classes modulo $p$ that do not contain zero."
– user7530
Sep 7 at 5:57
Yes, this is just a bit of sloppy wording. The intent is clearly "the set of equivalence classes modulo $p$ that do not contain zero."
– user7530
Sep 7 at 5:57
If you are going to argue language and syntax then you must consider what role exactly do "modulo n" serve in the phrase ""the (classes of) non-zero integers modulo n" It can't be an adjective because in is after the noun. It could refer to the classes modulo n but then how exactly was the "non-zero integers" cobbled in where it doesn't fit. To my mind an "[integer modulo n]" is a phrase we are suppose to recognize as a modulo class and so a "non-zero [integer modulo n]" is clearly a non-zero modulo class; of which there are two. Admittedly this makes the "class of integers" redundant.
– fleablood
Sep 7 at 6:09
If you are going to argue language and syntax then you must consider what role exactly do "modulo n" serve in the phrase ""the (classes of) non-zero integers modulo n" It can't be an adjective because in is after the noun. It could refer to the classes modulo n but then how exactly was the "non-zero integers" cobbled in where it doesn't fit. To my mind an "[integer modulo n]" is a phrase we are suppose to recognize as a modulo class and so a "non-zero [integer modulo n]" is clearly a non-zero modulo class; of which there are two. Admittedly this makes the "class of integers" redundant.
– fleablood
Sep 7 at 6:09
add a comment |Â
2 Answers
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up vote
6
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I think the "blocking" isn't and shouldn't be "[non-zero integers] [modulo $p$]" but rather "[non-zero][integers modulo $p$]"
You have three [integers modulo $3$]. They are $[53], [-216]$ and $[3,691]$. $[53]$ and $[3,691]$ are non-zero [integers modulo $3$]. And $[-216]$ is not a non-zero [integer modulo $3$].
answered Sep 7 at 6:17
fleablood
61.7k22678
What does it mean when you write two integers into the brackets?
– celtschk
Sep 7 at 6:33
@celtschk The comma supposedly is the thousends delimiter
– Hagen von Eitzen
Sep 7 at 6:35
@HagenvonEitzen: Thank you.
– celtschk
Sep 7 at 6:36
I would be fine with this interpretation, if the original formulation din't have the unfortunate "classes of" in the beginning
– Hagen von Eitzen
Sep 7 at 6:37
Thank you @fleablood. Just to make sure I understand, the example you gave is for a fixed $p = 3$, as I gave in the original comment, right?
– rb612
Sep 7 at 6:42
 |Â
show 4 more comments
up vote
5
down vote
Yes, interpretation 2 is the only one that makes sense. Why else would the term "non-zero" even be there?
An alternative phrasing with less ambiguity would be "non-zero classes of integers modulo $n$".
answered Sep 7 at 5:55
Arthur
102k797178
1
But that alternative is awkward. Nobody talks like that. The OP should recognize that interpretation 1 is essentially useless, so the OP has to get over the idea that anyone would ever care about interpretation 1. Many times in math there is an abuse of language, and this is one of those times.
– KCd
Sep 7 at 6:24
@KCd I must admit that I would have talked like that ...
– Hagen von Eitzen
Sep 7 at 6:38
@KCd you're right - I didn't ask this to be pedantic in any way, but rather, since these elementary concepts are so fundamental to my understanding of abstract algebra, I want to make sure that I myself am interpreting everything correctly, and that rather than me misunderstanding I wanted to get confirmation that there is a bit of language abuse going on here.
– rb612
Sep 7 at 6:40
I'd say "non-zero classes of integers modulo n". After all it is the classes not the integers that are non-zero (hence the entire reason for the OP). The only other option would be "classes of integers modulo n not congruent to zero" which is unnescessarily not. I suppose I'd actually so "non-zero classes modulo n" and assume the integers is implied.
– fleablood
Sep 7 at 21:26
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
I think the "blocking" isn't and shouldn't be "[non-zero integers] [modulo $p$]" but rather "[non-zero][integers modulo $p$]"
You have three [integers modulo $3$]. They are $[53], [-216]$ and $[3,691]$. $[53]$ and $[3,691]$ are non-zero [integers modulo $3$]. And $[-216]$ is not a non-zero [integer modulo $3$].
answered Sep 7 at 6:17
fleablood
61.7k22678
What does it mean when you write two integers into the brackets?
– celtschk
Sep 7 at 6:33
@celtschk The comma supposedly is the thousends delimiter
– Hagen von Eitzen
Sep 7 at 6:35
@HagenvonEitzen: Thank you.
– celtschk
Sep 7 at 6:36
I would be fine with this interpretation, if the original formulation din't have the unfortunate "classes of" in the beginning
– Hagen von Eitzen
Sep 7 at 6:37
Thank you @fleablood. Just to make sure I understand, the example you gave is for a fixed $p = 3$, as I gave in the original comment, right?
– rb612
Sep 7 at 6:42
 |Â
show 4 more comments
up vote
6
down vote
I think the "blocking" isn't and shouldn't be "[non-zero integers] [modulo $p$]" but rather "[non-zero][integers modulo $p$]"
You have three [integers modulo $3$]. They are $[53], [-216]$ and $[3,691]$. $[53]$ and $[3,691]$ are non-zero [integers modulo $3$]. And $[-216]$ is not a non-zero [integer modulo $3$].
answered Sep 7 at 6:17
fleablood
61.7k22678
What does it mean when you write two integers into the brackets?
– celtschk
Sep 7 at 6:33
@celtschk The comma supposedly is the thousends delimiter
– Hagen von Eitzen
Sep 7 at 6:35
@HagenvonEitzen: Thank you.
– celtschk
Sep 7 at 6:36
I would be fine with this interpretation, if the original formulation din't have the unfortunate "classes of" in the beginning
– Hagen von Eitzen
Sep 7 at 6:37
Thank you @fleablood. Just to make sure I understand, the example you gave is for a fixed $p = 3$, as I gave in the original comment, right?
– rb612
Sep 7 at 6:42
 |Â
show 4 more comments
up vote
6
down vote
up vote
6
down vote
I think the "blocking" isn't and shouldn't be "[non-zero integers] [modulo $p$]" but rather "[non-zero][integers modulo $p$]"
You have three [integers modulo $3$]. They are $[53], [-216]$ and $[3,691]$. $[53]$ and $[3,691]$ are non-zero [integers modulo $3$]. And $[-216]$ is not a non-zero [integer modulo $3$].
answered Sep 7 at 6:17
fleablood
61.7k22678
I think the "blocking" isn't and shouldn't be "[non-zero integers] [modulo $p$]" but rather "[non-zero][integers modulo $p$]"
You have three [integers modulo $3$]. They are $[53], [-216]$ and $[3,691]$. $[53]$ and $[3,691]$ are non-zero [integers modulo $3$]. And $[-216]$ is not a non-zero [integer modulo $3$].
answered Sep 7 at 6:17
fleablood
61.7k22678
edited Sep 7 at 16:16
answered Sep 7 at 6:17
fleablood
61.7k22678
answered Sep 7 at 6:17
fleablood
61.7k22678
answered Sep 7 at 6:17
fleablood
61.7k22678
61.7k22678
What does it mean when you write two integers into the brackets?
– celtschk
Sep 7 at 6:33
@celtschk The comma supposedly is the thousends delimiter
– Hagen von Eitzen
Sep 7 at 6:35
@HagenvonEitzen: Thank you.
– celtschk
Sep 7 at 6:36
I would be fine with this interpretation, if the original formulation din't have the unfortunate "classes of" in the beginning
– Hagen von Eitzen
Sep 7 at 6:37
Thank you @fleablood. Just to make sure I understand, the example you gave is for a fixed $p = 3$, as I gave in the original comment, right?
– rb612
Sep 7 at 6:42
 |Â
show 4 more comments
What does it mean when you write two integers into the brackets?
– celtschk
Sep 7 at 6:33
@celtschk The comma supposedly is the thousends delimiter
– Hagen von Eitzen
Sep 7 at 6:35
@HagenvonEitzen: Thank you.
– celtschk
Sep 7 at 6:36
I would be fine with this interpretation, if the original formulation din't have the unfortunate "classes of" in the beginning
– Hagen von Eitzen
Sep 7 at 6:37
Thank you @fleablood. Just to make sure I understand, the example you gave is for a fixed $p = 3$, as I gave in the original comment, right?
– rb612
Sep 7 at 6:42
What does it mean when you write two integers into the brackets?
– celtschk
Sep 7 at 6:33
What does it mean when you write two integers into the brackets?
– celtschk
Sep 7 at 6:33
@celtschk The comma supposedly is the thousends delimiter
– Hagen von Eitzen
Sep 7 at 6:35
@celtschk The comma supposedly is the thousends delimiter
– Hagen von Eitzen
Sep 7 at 6:35
@HagenvonEitzen: Thank you.
– celtschk
Sep 7 at 6:36
@HagenvonEitzen: Thank you.
– celtschk
Sep 7 at 6:36
I would be fine with this interpretation, if the original formulation din't have the unfortunate "classes of" in the beginning
– Hagen von Eitzen
Sep 7 at 6:37
I would be fine with this interpretation, if the original formulation din't have the unfortunate "classes of" in the beginning
– Hagen von Eitzen
Sep 7 at 6:37
Thank you @fleablood. Just to make sure I understand, the example you gave is for a fixed $p = 3$, as I gave in the original comment, right?
– rb612
Sep 7 at 6:42
Thank you @fleablood. Just to make sure I understand, the example you gave is for a fixed $p = 3$, as I gave in the original comment, right?
– rb612
Sep 7 at 6:42
 |Â
show 4 more comments
up vote
5
down vote
Yes, interpretation 2 is the only one that makes sense. Why else would the term "non-zero" even be there?
An alternative phrasing with less ambiguity would be "non-zero classes of integers modulo $n$".
answered Sep 7 at 5:55
Arthur
102k797178
1
But that alternative is awkward. Nobody talks like that. The OP should recognize that interpretation 1 is essentially useless, so the OP has to get over the idea that anyone would ever care about interpretation 1. Many times in math there is an abuse of language, and this is one of those times.
– KCd
Sep 7 at 6:24
@KCd I must admit that I would have talked like that ...
– Hagen von Eitzen
Sep 7 at 6:38
@KCd you're right - I didn't ask this to be pedantic in any way, but rather, since these elementary concepts are so fundamental to my understanding of abstract algebra, I want to make sure that I myself am interpreting everything correctly, and that rather than me misunderstanding I wanted to get confirmation that there is a bit of language abuse going on here.
– rb612
Sep 7 at 6:40
I'd say "non-zero classes of integers modulo n". After all it is the classes not the integers that are non-zero (hence the entire reason for the OP). The only other option would be "classes of integers modulo n not congruent to zero" which is unnescessarily not. I suppose I'd actually so "non-zero classes modulo n" and assume the integers is implied.
– fleablood
Sep 7 at 21:26
add a comment |Â
up vote
5
down vote
Yes, interpretation 2 is the only one that makes sense. Why else would the term "non-zero" even be there?
An alternative phrasing with less ambiguity would be "non-zero classes of integers modulo $n$".
answered Sep 7 at 5:55
Arthur
102k797178
1
But that alternative is awkward. Nobody talks like that. The OP should recognize that interpretation 1 is essentially useless, so the OP has to get over the idea that anyone would ever care about interpretation 1. Many times in math there is an abuse of language, and this is one of those times.
– KCd
Sep 7 at 6:24
@KCd I must admit that I would have talked like that ...
– Hagen von Eitzen
Sep 7 at 6:38
@KCd you're right - I didn't ask this to be pedantic in any way, but rather, since these elementary concepts are so fundamental to my understanding of abstract algebra, I want to make sure that I myself am interpreting everything correctly, and that rather than me misunderstanding I wanted to get confirmation that there is a bit of language abuse going on here.
– rb612
Sep 7 at 6:40
I'd say "non-zero classes of integers modulo n". After all it is the classes not the integers that are non-zero (hence the entire reason for the OP). The only other option would be "classes of integers modulo n not congruent to zero" which is unnescessarily not. I suppose I'd actually so "non-zero classes modulo n" and assume the integers is implied.
– fleablood
Sep 7 at 21:26
add a comment |Â
up vote
5
down vote
up vote
5
down vote
Yes, interpretation 2 is the only one that makes sense. Why else would the term "non-zero" even be there?
An alternative phrasing with less ambiguity would be "non-zero classes of integers modulo $n$".
answered Sep 7 at 5:55
Arthur
102k797178
Yes, interpretation 2 is the only one that makes sense. Why else would the term "non-zero" even be there?
An alternative phrasing with less ambiguity would be "non-zero classes of integers modulo $n$".
answered Sep 7 at 5:55
Arthur
102k797178
answered Sep 7 at 5:55
Arthur
102k797178
answered Sep 7 at 5:55
Arthur
102k797178
answered Sep 7 at 5:55
Arthur
102k797178
102k797178
1
But that alternative is awkward. Nobody talks like that. The OP should recognize that interpretation 1 is essentially useless, so the OP has to get over the idea that anyone would ever care about interpretation 1. Many times in math there is an abuse of language, and this is one of those times.
– KCd
Sep 7 at 6:24
@KCd I must admit that I would have talked like that ...
– Hagen von Eitzen
Sep 7 at 6:38
@KCd you're right - I didn't ask this to be pedantic in any way, but rather, since these elementary concepts are so fundamental to my understanding of abstract algebra, I want to make sure that I myself am interpreting everything correctly, and that rather than me misunderstanding I wanted to get confirmation that there is a bit of language abuse going on here.
– rb612
Sep 7 at 6:40
I'd say "non-zero classes of integers modulo n". After all it is the classes not the integers that are non-zero (hence the entire reason for the OP). The only other option would be "classes of integers modulo n not congruent to zero" which is unnescessarily not. I suppose I'd actually so "non-zero classes modulo n" and assume the integers is implied.
– fleablood
Sep 7 at 21:26
add a comment |Â
1
But that alternative is awkward. Nobody talks like that. The OP should recognize that interpretation 1 is essentially useless, so the OP has to get over the idea that anyone would ever care about interpretation 1. Many times in math there is an abuse of language, and this is one of those times.
– KCd
Sep 7 at 6:24
@KCd I must admit that I would have talked like that ...
– Hagen von Eitzen
Sep 7 at 6:38
@KCd you're right - I didn't ask this to be pedantic in any way, but rather, since these elementary concepts are so fundamental to my understanding of abstract algebra, I want to make sure that I myself am interpreting everything correctly, and that rather than me misunderstanding I wanted to get confirmation that there is a bit of language abuse going on here.
– rb612
Sep 7 at 6:40
I'd say "non-zero classes of integers modulo n". After all it is the classes not the integers that are non-zero (hence the entire reason for the OP). The only other option would be "classes of integers modulo n not congruent to zero" which is unnescessarily not. I suppose I'd actually so "non-zero classes modulo n" and assume the integers is implied.
– fleablood
Sep 7 at 21:26
1
1
But that alternative is awkward. Nobody talks like that. The OP should recognize that interpretation 1 is essentially useless, so the OP has to get over the idea that anyone would ever care about interpretation 1. Many times in math there is an abuse of language, and this is one of those times.
– KCd
Sep 7 at 6:24
But that alternative is awkward. Nobody talks like that. The OP should recognize that interpretation 1 is essentially useless, so the OP has to get over the idea that anyone would ever care about interpretation 1. Many times in math there is an abuse of language, and this is one of those times.
– KCd
Sep 7 at 6:24
@KCd I must admit that I would have talked like that ...
– Hagen von Eitzen
Sep 7 at 6:38
@KCd I must admit that I would have talked like that ...
– Hagen von Eitzen
Sep 7 at 6:38
@KCd you're right - I didn't ask this to be pedantic in any way, but rather, since these elementary concepts are so fundamental to my understanding of abstract algebra, I want to make sure that I myself am interpreting everything correctly, and that rather than me misunderstanding I wanted to get confirmation that there is a bit of language abuse going on here.
– rb612
Sep 7 at 6:40
@KCd you're right - I didn't ask this to be pedantic in any way, but rather, since these elementary concepts are so fundamental to my understanding of abstract algebra, I want to make sure that I myself am interpreting everything correctly, and that rather than me misunderstanding I wanted to get confirmation that there is a bit of language abuse going on here.
– rb612
Sep 7 at 6:40
I'd say "non-zero classes of integers modulo n". After all it is the classes not the integers that are non-zero (hence the entire reason for the OP). The only other option would be "classes of integers modulo n not congruent to zero" which is unnescessarily not. I suppose I'd actually so "non-zero classes modulo n" and assume the integers is implied.
– fleablood
Sep 7 at 21:26
I'd say "non-zero classes of integers modulo n". After all it is the classes not the integers that are non-zero (hence the entire reason for the OP). The only other option would be "classes of integers modulo n not congruent to zero" which is unnescessarily not. I suppose I'd actually so "non-zero classes modulo n" and assume the integers is implied.
– fleablood
Sep 7 at 21:26
add a comment |Â
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1
Yes, this is just a bit of sloppy wording. The intent is clearly "the set of equivalence classes modulo $p$ that do not contain zero."
– user7530
Sep 7 at 5:57
If you are going to argue language and syntax then you must consider what role exactly do "modulo n" serve in the phrase ""the (classes of) non-zero integers modulo n" It can't be an adjective because in is after the noun. It could refer to the classes modulo n but then how exactly was the "non-zero integers" cobbled in where it doesn't fit. To my mind an "[integer modulo n]" is a phrase we are suppose to recognize as a modulo class and so a "non-zero [integer modulo n]" is clearly a non-zero modulo class; of which there are two. Admittedly this makes the "class of integers" redundant.
– fleablood
Sep 7 at 6:09