Question

Non Zero ideals of $\mathbf{Z}_{12}$

score 303 · Answer 1 · 2026-03-27 23:31:15

303

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Non Zero ideals of $\mathbf{Z}_{12}$

Published on 27 Mar 2026 - 23:31

score 1k · Answer 2 · 2018-06-30 08:37:00

1k

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Proving every prime ideal is maximal in a finite dimensional ring over a field.

Published on 30 Jun 2018 - 8:37

score 129 · Answer 3 · 2018-06-30 16:56:25

Equivalence class $r$ in $R$ is represented by $r=a_0 + a_1 X + \cdots+ a_{n-1} X^{n-1},$ with $a_i \in K.$ Show $r \in R$ is a unit iff $a_0 \neq 0$.

score 181 · Answer 4 · 2018-06-30 20:50:14

181

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Finding a ring isomorphism $K [X] / ((X-a)^n) \cong K [X] / (X^n)$ for every $a \in K.$

Published on 30 Jun 2018 - 20:50

score 138 · Answer 5 · 2026-03-22 17:12:58

138

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Subring of $\text{Mat}_n(Z_m)$ is commutative if $x^2=0 \implies x=0$.

Published on 22 Mar 2026 - 17:12

score 143 · Answer 6 · 2018-07-01 22:49:24

143

Views

Factoring the polynomial $2X^{10}+4X^5 +3$ into irreducible factors in $(\mathbb{Z} / 5\mathbb{Z})[X].$

Published on 01 Jul 2018 - 22:49

score 137 · Answer 7 · 2026-03-28 22:31:07

137

Views

If $R$ is a commutative ring with ideals $A$ and $B$ such that $AB$ is principal, then $A$ and $B$ are finitely generated.

Published on 28 Mar 2026 - 22:31

score 613 · Answer 8 · 2018-07-02 10:05:17

613

Views

Associates of greatest common divisor in UFD

Published on 02 Jul 2018 - 10:05

score 66 · Answer 9 · 2026-03-25 12:53:18

66

Views

Irreductibility of a polynom

Published on 25 Mar 2026 - 12:53

score 21 · Answer 10 · 2018-07-02 16:55:56

21

Views

Show that $f=\sum_{j=0}^{n}f_jx^j$ for $f\in R[x]$ and $f_i=0$ for all $i>n$.

Published on 02 Jul 2018 - 16:55

score 103 · Answer 11 · 2026-03-28 22:31:00

103

Views

Prove that the ideal of $p\in \mathbb Z[x]$ with $p(1)$ even is not principal

Published on 28 Mar 2026 - 22:31

score 193 · Answer 12 · 2018-07-02 19:26:51

193

Views

Finding homomorphism between rings

Published on 02 Jul 2018 - 19:26

score 201 · Answer 13 · 2026-03-25 14:25:54

201

Views

Proving that the polynomial $10X^6-15X^2+7$ is irreducible in $\mathbb{Q}[X].$

Published on 25 Mar 2026 - 14:25

score 154 · Answer 14 · 2026-03-29 03:36:09

154

Views

Connection between ideals in $R$ and those in $R[x]$

Published on 29 Mar 2026 - 3:36

score 106 · Answer 15 · 2026-03-25 16:08:17

106

Views

$A\times B\simeq C\times D \implies A\simeq C\text{ or } A\simeq D$

Published on 25 Mar 2026 - 16:08

List Question

Non Zero ideals of $\mathbf{Z}_{12}$

Proving every prime ideal is maximal in a finite dimensional ring over a field.

Equivalence class $r$ in $R$ is represented by $r=a_0 + a_1 X + \cdots+ a_{n-1} X^{n-1},$ with $a_i \in K.$ Show $r \in R$ is a unit iff $a_0 \neq 0$.

Finding a ring isomorphism $K [X] / ((X-a)^n) \cong K [X] / (X^n)$ for every $a \in K.$

Subring of $\text{Mat}_n(Z_m)$ is commutative if $x^2=0 \implies x=0$.

Factoring the polynomial $2X^{10}+4X^5 +3$ into irreducible factors in $(\mathbb{Z} / 5\mathbb{Z})[X].$

If $R$ is a commutative ring with ideals $A$ and $B$ such that $AB$ is principal, then $A$ and $B$ are finitely generated.

Associates of greatest common divisor in UFD

Irreductibility of a polynom

Show that $f=\sum_{j=0}^{n}f_jx^j$ for $f\in R[x]$ and $f_i=0$ for all $i>n$.

Prove that the ideal of $p\in \mathbb Z[x]$ with $p(1)$ even is not principal

Finding homomorphism between rings

Proving that the polynomial $10X^6-15X^2+7$ is irreducible in $\mathbb{Q}[X].$

Connection between ideals in $R$ and those in $R[x]$

$A\times B\simeq C\times D \implies A\simeq C\text{ or } A\simeq D$

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