Question

Ideal of factor ring

score 39 · Answer 1 · 2019-03-12 02:51:25

39

Views

Ideal of factor ring

Published on 12 Mar 2019 - 2:51

score 1.4k · Answer 2 · 2019-03-12 08:17:29

1.4k

Views

Let $R$ be a domain. Prove that if a polynomial in $R[x]$ is a unit, then it is a nonzero constant (the converse is true if $R$ is a field)

Published on 12 Mar 2019 - 8:17

score 95 · Answer 3 · 2019-03-12 11:52:30

95

Views

$(A,+)$ and $(U(A), \cdot)$ are not isomorphic

Published on 12 Mar 2019 - 11:52

score 79 · Answer 4 · 2019-03-12 13:14:04

79

Views

$x^2=x\iff x=0\lor x=1$. If $a,b\in A$ s.t. $ab=a+b\to ab=ba$

Published on 12 Mar 2019 - 13:14

score 41 · Answer 5 · 2019-03-12 16:35:35

41

Views

Dividing $x^3+2$ by $2x^2+3x+4$ in $\mathbb{Z}[x]$

Published on 12 Mar 2019 - 16:35

score 121 · Answer 6 · 2019-03-12 17:55:19

121

Views

Maximum order of a polynomial permutation of a finite ring

Published on 12 Mar 2019 - 17:55

score 472 · Answer 7 · 2026-04-06 00:47:19

472

Views

Is it possible to have an Abelian group under two different binary operations but the binary operations are not distributive?

Published on 06 Apr 2026 - 0:47

score 65 · Answer 8 · 2026-04-06 00:43:14

65

Views

Fields and Groups equivalent

Published on 06 Apr 2026 - 0:43

score 286 · Answer 9 · 2026-03-30 15:00:36

286

Views

Homogeneous elements of an ideal over a quotient ring

Published on 30 Mar 2026 - 15:00

score 522 · Answer 10 · 2026-03-31 12:17:14

522

Views

Frac$(R)=F$, quotient field of the integral domain $R$, then Frac$(R[x]) \cong F(x)$ and also Frac$(R[x]) \cong F(x_{1},x_{2},....,x_{n})$.

Published on 31 Mar 2026 - 12:17

score 23 · Answer 11 · 2019-03-12 22:30:21

23

Views

Consider $R=k[x_1,x_2,...x_n]$. Difference between $(f_1,..,f_k)$ and $k[f_1,...f_k]$

Published on 12 Mar 2019 - 22:30

score 515 · Answer 12 · 2019-03-12 22:58:08

515

Views

Ideal generated by homogeneous polynomials

Published on 12 Mar 2019 - 22:58

score 1.2k · Answer 13 · 2026-04-06 00:50:39

1.2k

Views

Let $F$ be a field. (a) If $1 + 1 = 0$, show that $a + a = 0$ for all $a \in F$. (b) If $a + a = 0$ for some $a \neq 0$, show that $1 + 1 = 0$

Published on 06 Apr 2026 - 0:50

score 299 · Answer 14 · 2026-03-25 09:38:40

299

Views

Why $\mathbb{Z}[\sqrt{-5}]\cong\mathbb{Z}[X]/(X^2+5)$ and $\mathbb{C}\cong \mathbb{R}[X]/(X^2+1)$?

Published on 25 Mar 2026 - 9:38

score 53 · Answer 15 · 2026-03-25 04:59:04

53

Views

If $B$ is an $A$-algebra of Noetherian rings then $\text{Ass}_{A}M=\{f^{-1}(p):p \in \text{Ass}_{B}M\}$ for a f.g. $B$-module $M.$

Published on 25 Mar 2026 - 4:59

List Question

Ideal of factor ring

Let $R$ be a domain. Prove that if a polynomial in $R[x]$ is a unit, then it is a nonzero constant (the converse is true if $R$ is a field)

$(A,+)$ and $(U(A), \cdot)$ are not isomorphic

$x^2=x\iff x=0\lor x=1$. If $a,b\in A$ s.t. $ab=a+b\to ab=ba$

Dividing $x^3+2$ by $2x^2+3x+4$ in $\mathbb{Z}[x]$

Maximum order of a polynomial permutation of a finite ring

Is it possible to have an Abelian group under two different binary operations but the binary operations are not distributive?

Fields and Groups equivalent

Homogeneous elements of an ideal over a quotient ring

Frac$(R)=F$, quotient field of the integral domain $R$, then Frac$(R[x]) \cong F(x)$ and also Frac$(R[x]) \cong F(x_{1},x_{2},....,x_{n})$.

Consider $R=k[x_1,x_2,...x_n]$. Difference between $(f_1,..,f_k)$ and $k[f_1,...f_k]$

Ideal generated by homogeneous polynomials

Let $F$ be a field. (a) If $1 + 1 = 0$, show that $a + a = 0$ for all $a \in F$. (b) If $a + a = 0$ for some $a \neq 0$, show that $1 + 1 = 0$

Why $\mathbb{Z}[\sqrt{-5}]\cong\mathbb{Z}[X]/(X^2+5)$ and $\mathbb{C}\cong \mathbb{R}[X]/(X^2+1)$?

If $B$ is an $A$-algebra of Noetherian rings then $\text{Ass}_{A}M=\{f^{-1}(p):p \in \text{Ass}_{B}M\}$ for a f.g. $B$-module $M.$

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