List questions from maximal-and-prime-ideals

Published on 06 Mar 2014 - 12:12

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Ideals contained in the exactly same prime ideals

Published on 09 Apr 2017 - 17:55

1.4k

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Show that $M = \{a + b\sqrt{2}\mid 5\mid a, 5 \mid b\}$ is a maximal ideal in $R = \mathbb{Z}[\sqrt{2}]$ and find $R/M$.

Published on 10 Apr 2017 - 11:17

820

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$I:=\{f(x)\in R\mid f(1)=0\}$ is a maximal ideal?

Published on 12 Apr 2017 - 16:37

204

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If $A$ is a prime ideal on $R\times S$, then $A = P \times S$, where P is a prime ideal on R or $A = R\times Q$, where $Q$ is a prime ideal on $S$.

Published on 15 Apr 2017 - 18:59

212

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Ring homomorphism and prime ideals

Published on 16 Apr 2017 - 14:22

1.3k

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Is $\langle x^2+1\rangle$ a maximal ideal in $Z[x]$? What about $\langle x^2+1,5\rangle$?

Published on 16 Apr 2017 - 23:46

261

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Counter example involving primary ideals (Atiyah-Mac Donald)

Published on 17 Apr 2017 - 13:57

136

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Under What Conditions is an ideal $I \subset R$ in the given ring $R$ prime?

Published on 17 Apr 2017 - 16:00

73

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For $R = R_{1} \oplus R_{2}$, where $I_{1}$ ideal of $R_{1}$, $I_{2}$ ideal of $R_{2}$, show that $I_{1},I_{2}$ ideals of $R$

Published on 30 Mar 2026 - 8:37

96

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About Spec of ring

Published on 19 Apr 2017 - 14:16

277

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Confusion about primary decomposition

Published on 26 Mar 2026 - 19:15

57

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Showing that a subset of a ring with a certain property is a union of prime ideals.

Published on 23 Apr 2017 - 18:11

787

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Given an Artinian ring $(A, \mathfrak{m})$, show that $\mathfrak{m}$ is nilpotent.

Published on 25 Mar 2026 - 17:24

306

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To show a certain ideal is maximal.

Published on 26 Apr 2017 - 21:10

List Question

Ideal in a ring of continuous functions

Ideals contained in the exactly same prime ideals

Show that $M = \{a + b\sqrt{2}\mid 5\mid a, 5 \mid b\}$ is a maximal ideal in $R = \mathbb{Z}[\sqrt{2}]$ and find $R/M$.

$I:=\{f(x)\in R\mid f(1)=0\}$ is a maximal ideal?

If $A$ is a prime ideal on $R\times S$, then $A = P \times S$, where P is a prime ideal on R or $A = R\times Q$, where $Q$ is a prime ideal on $S$.

Ring homomorphism and prime ideals

Is $\langle x^2+1\rangle$ a maximal ideal in $Z[x]$? What about $\langle x^2+1,5\rangle$?

Counter example involving primary ideals (Atiyah-Mac Donald)

Under What Conditions is an ideal $I \subset R$ in the given ring $R$ prime?

For $R = R_{1} \oplus R_{2}$, where $I_{1}$ ideal of $R_{1}$, $I_{2}$ ideal of $R_{2}$, show that $I_{1},I_{2}$ ideals of $R$

About Spec of ring

Confusion about primary decomposition

Showing that a subset of a ring with a certain property is a union of prime ideals.

Given an Artinian ring $(A, \mathfrak{m})$, show that $\mathfrak{m}$ is nilpotent.

To show a certain ideal is maximal.

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