Relation between Jacobson radical of an ideal and the jacobson radical of the ring
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Let $R$ be a ring with $1$, and $I$ and ideal of $R$. Let $P$ be a prime ideal of $I$. Is it true that $rad(I/P) subseteq rad(R/P)$? I am really confused if this inclusion is true or not.
abstract-algebra ring-theory maximal-and-prime-ideals
asked Nov 29 at 3:47
P.G
301110
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up vote
2
down vote
favorite
Let $R$ be a ring with $1$, and $I$ and ideal of $R$. Let $P$ be a prime ideal of $I$. Is it true that $rad(I/P) subseteq rad(R/P)$? I am really confused if this inclusion is true or not.
abstract-algebra ring-theory maximal-and-prime-ideals
asked Nov 29 at 3:47
P.G
301110
Well what do you want to ask about, the definition or the inclusion? What is puzzling you about the definition?
– rschwieb
Nov 29 at 14:38
I am confusing about the inclusion. I don’t know how to show this inclusion trough definition, so I don’t know if the inclusion its true. Or maybe I don’t understand the definition. Is this inclusion true? I will edit the question.
– P.G
Nov 29 at 19:48
Intuitively I would expect it to be false: it seems like the maximal ideals of $I/P$ would probably be different animals than those of $R/P$.
– rschwieb
Nov 29 at 20:08
Yes, I thought that too. I am studying this article core.ac.uk/download/pdf/82204709.pdf . In Proposition 1 the author says ''... In this quotient the prime radical coincides with the Jacobson radical, which contains J/P. Thus J/P is a lower nil radical ring '' I don't know how the author concludes that J/P is a lower nil radical ring. Is it true that if a ring $R/P$ is a Jacobson ring, then for every ring $I/P$ in $R/P$, both radicals coincides in $I/P$?. If so, why? Thank you.
– P.G
Nov 29 at 20:17
I take it back: I didn't realize the paper was talking about noncommutative Jacobson rings. Necessarily, we have to use rings without identity. I see that the argument does rely on the step you are talking about. Perhaps it appears in Jacobson's basic texts discussion the radical?
– rschwieb
Nov 30 at 14:47
|
show 5 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $R$ be a ring with $1$, and $I$ and ideal of $R$. Let $P$ be a prime ideal of $I$. Is it true that $rad(I/P) subseteq rad(R/P)$? I am really confused if this inclusion is true or not.
abstract-algebra ring-theory maximal-and-prime-ideals
asked Nov 29 at 3:47
P.G
301110
Let $R$ be a ring with $1$, and $I$ and ideal of $R$. Let $P$ be a prime ideal of $I$. Is it true that $rad(I/P) subseteq rad(R/P)$? I am really confused if this inclusion is true or not.
abstract-algebra ring-theory maximal-and-prime-ideals
abstract-algebra ring-theory maximal-and-prime-ideals
asked Nov 29 at 3:47
P.G
301110
asked Nov 29 at 3:47
P.G
301110
edited Nov 29 at 19:49
asked Nov 29 at 3:47
P.G
301110
asked Nov 29 at 3:47
P.G
301110
asked Nov 29 at 3:47
P.G
301110
301110
Well what do you want to ask about, the definition or the inclusion? What is puzzling you about the definition?
– rschwieb
Nov 29 at 14:38
I am confusing about the inclusion. I don’t know how to show this inclusion trough definition, so I don’t know if the inclusion its true. Or maybe I don’t understand the definition. Is this inclusion true? I will edit the question.
– P.G
Nov 29 at 19:48
Intuitively I would expect it to be false: it seems like the maximal ideals of $I/P$ would probably be different animals than those of $R/P$.
– rschwieb
Nov 29 at 20:08
Yes, I thought that too. I am studying this article core.ac.uk/download/pdf/82204709.pdf . In Proposition 1 the author says ''... In this quotient the prime radical coincides with the Jacobson radical, which contains J/P. Thus J/P is a lower nil radical ring '' I don't know how the author concludes that J/P is a lower nil radical ring. Is it true that if a ring $R/P$ is a Jacobson ring, then for every ring $I/P$ in $R/P$, both radicals coincides in $I/P$?. If so, why? Thank you.
– P.G
Nov 29 at 20:17
I take it back: I didn't realize the paper was talking about noncommutative Jacobson rings. Necessarily, we have to use rings without identity. I see that the argument does rely on the step you are talking about. Perhaps it appears in Jacobson's basic texts discussion the radical?
– rschwieb
Nov 30 at 14:47
|
show 5 more comments
Well what do you want to ask about, the definition or the inclusion? What is puzzling you about the definition?
– rschwieb
Nov 29 at 14:38
I am confusing about the inclusion. I don’t know how to show this inclusion trough definition, so I don’t know if the inclusion its true. Or maybe I don’t understand the definition. Is this inclusion true? I will edit the question.
– P.G
Nov 29 at 19:48
Intuitively I would expect it to be false: it seems like the maximal ideals of $I/P$ would probably be different animals than those of $R/P$.
– rschwieb
Nov 29 at 20:08
Yes, I thought that too. I am studying this article core.ac.uk/download/pdf/82204709.pdf . In Proposition 1 the author says ''... In this quotient the prime radical coincides with the Jacobson radical, which contains J/P. Thus J/P is a lower nil radical ring '' I don't know how the author concludes that J/P is a lower nil radical ring. Is it true that if a ring $R/P$ is a Jacobson ring, then for every ring $I/P$ in $R/P$, both radicals coincides in $I/P$?. If so, why? Thank you.
– P.G
Nov 29 at 20:17
I take it back: I didn't realize the paper was talking about noncommutative Jacobson rings. Necessarily, we have to use rings without identity. I see that the argument does rely on the step you are talking about. Perhaps it appears in Jacobson's basic texts discussion the radical?
– rschwieb
Nov 30 at 14:47
Well what do you want to ask about, the definition or the inclusion? What is puzzling you about the definition?
– rschwieb
Nov 29 at 14:38
Well what do you want to ask about, the definition or the inclusion? What is puzzling you about the definition?
– rschwieb
Nov 29 at 14:38
I am confusing about the inclusion. I don’t know how to show this inclusion trough definition, so I don’t know if the inclusion its true. Or maybe I don’t understand the definition. Is this inclusion true? I will edit the question.
– P.G
Nov 29 at 19:48
I am confusing about the inclusion. I don’t know how to show this inclusion trough definition, so I don’t know if the inclusion its true. Or maybe I don’t understand the definition. Is this inclusion true? I will edit the question.
– P.G
Nov 29 at 19:48
Intuitively I would expect it to be false: it seems like the maximal ideals of $I/P$ would probably be different animals than those of $R/P$.
– rschwieb
Nov 29 at 20:08
Intuitively I would expect it to be false: it seems like the maximal ideals of $I/P$ would probably be different animals than those of $R/P$.
– rschwieb
Nov 29 at 20:08
Yes, I thought that too. I am studying this article core.ac.uk/download/pdf/82204709.pdf . In Proposition 1 the author says ''... In this quotient the prime radical coincides with the Jacobson radical, which contains J/P. Thus J/P is a lower nil radical ring '' I don't know how the author concludes that J/P is a lower nil radical ring. Is it true that if a ring $R/P$ is a Jacobson ring, then for every ring $I/P$ in $R/P$, both radicals coincides in $I/P$?. If so, why? Thank you.
– P.G
Nov 29 at 20:17
Yes, I thought that too. I am studying this article core.ac.uk/download/pdf/82204709.pdf . In Proposition 1 the author says ''... In this quotient the prime radical coincides with the Jacobson radical, which contains J/P. Thus J/P is a lower nil radical ring '' I don't know how the author concludes that J/P is a lower nil radical ring. Is it true that if a ring $R/P$ is a Jacobson ring, then for every ring $I/P$ in $R/P$, both radicals coincides in $I/P$?. If so, why? Thank you.
– P.G
Nov 29 at 20:17
I take it back: I didn't realize the paper was talking about noncommutative Jacobson rings. Necessarily, we have to use rings without identity. I see that the argument does rely on the step you are talking about. Perhaps it appears in Jacobson's basic texts discussion the radical?
– rschwieb
Nov 30 at 14:47
I take it back: I didn't realize the paper was talking about noncommutative Jacobson rings. Necessarily, we have to use rings without identity. I see that the argument does rely on the step you are talking about. Perhaps it appears in Jacobson's basic texts discussion the radical?
– rschwieb
Nov 30 at 14:47
|
show 5 more comments
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Well what do you want to ask about, the definition or the inclusion? What is puzzling you about the definition?
– rschwieb
Nov 29 at 14:38
I am confusing about the inclusion. I don’t know how to show this inclusion trough definition, so I don’t know if the inclusion its true. Or maybe I don’t understand the definition. Is this inclusion true? I will edit the question.
– P.G
Nov 29 at 19:48
Intuitively I would expect it to be false: it seems like the maximal ideals of $I/P$ would probably be different animals than those of $R/P$.
– rschwieb
Nov 29 at 20:08
Yes, I thought that too. I am studying this article core.ac.uk/download/pdf/82204709.pdf . In Proposition 1 the author says ''... In this quotient the prime radical coincides with the Jacobson radical, which contains J/P. Thus J/P is a lower nil radical ring '' I don't know how the author concludes that J/P is a lower nil radical ring. Is it true that if a ring $R/P$ is a Jacobson ring, then for every ring $I/P$ in $R/P$, both radicals coincides in $I/P$?. If so, why? Thank you.
– P.G
Nov 29 at 20:17
I take it back: I didn't realize the paper was talking about noncommutative Jacobson rings. Necessarily, we have to use rings without identity. I see that the argument does rely on the step you are talking about. Perhaps it appears in Jacobson's basic texts discussion the radical?
– rschwieb
Nov 30 at 14:47