User user138017

2

12

votes

3

answers

456

views

Suppose $e^A = A$, prove that $A$ is diagonalizable

matrices diagonalization

Jan 29 '16 at 15:59 user60589 2,475

2

9

votes

2

answers

582

views

Is there always an intermediate field between non-prime field extension?

abstract-algebra field-theory galois-theory extension-field

Aug 16 '16 at 10:51 Jyrki Lahtonen 118k

9

8

votes

1

answer

4k

views

if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian [duplicate]

abstract-algebra algebraic-geometry ring-theory commutative-algebra ideals

Feb 2 '15 at 12:53 user 1 6,691

1

6

votes

1

answer

145

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If $A, B$ are positive definite and $\|A+B\| = \|A\|+\|B\|$, they have a common eigenvector

linear-algebra matrices eigenvalues-eigenvectors norm

Dec 29 '16 at 10:58 user1551 107k

2

6

votes

2

answers

3k

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Show the projective space $RP^2$ minus a point is homotopy equivalent to the unit circle $S^1$

general-topology algebraic-topology fundamental-groups

Apr 3 '15 at 19:39 Anubhav Mukherjee 5,833

1

6

votes

1

answer

133

views

A number related to the roots of a quartic polynomial is a root of a cubic polynomial

algebra-precalculus polynomials roots

Jul 25 '14 at 1:44 Community♦ 1

1

5

votes

1

answer

141

views

Show an ideal is a finitely generated projective module via a split exact sequence

abstract-algebra commutative-algebra modules tensor-products exact-sequence

Mar 13 '15 at 15:14 user26857 42.7k

3

votes

1

answer

241

views

orthogonal matrix with negative determinant

linear-algebra matrices

May 1 '18 at 20:41 user138017 991

1

3

votes

1

answer

129

views

$A,B$ are orthogonal projections and $\|Ax\|^2+\|Bx\|^2=\|x\|^2$ show $A+B=I$

linear-algebra matrices inner-products projection-matrices

Apr 23 '18 at 22:20 user138017 991

2

votes

2

answers

643

views

Exercise 3.40 from Folland, Real Analysis

calculus real-analysis analysis measure-theory derivatives

Jan 25 '16 at 21:46 Lucas 1,186

2

votes

1

answer

202

views

How to use Seifert–van Kampen to find $\pi_1( RP^n ∨ RP^n)$

abstract-algebra general-topology algebraic-topology fundamental-groups

Apr 9 '15 at 8:06 Tony 5,978

2

votes

2

answers

49

views

Show $(⊕M_α)/I(⊕M_α)\simeq ⊕(M_α/IM_α)$ for ${M_α}$ $R$-modules, and $I$ is an ideal of the commutative ring $R$

abstract-algebra modules ideals direct-sum

Feb 23 '15 at 9:25 user26857 42.7k

2

votes

1

answer

166

views

homotopic closed paths in topological group

group-theory algebraic-topology homotopy-theory fundamental-groups

Feb 17 '15 at 2:46 kobe 37.7k

2

votes

2

answers

157

views

The finitely generated-ness of ideals $I +rR$ and $I:r$ imply $I$ is a finitely generated ideal [closed]

abstract-algebra ring-theory commutative-algebra ideals finitely-generated

Feb 2 '15 at 14:16 user26857 42.7k

2

votes

2

answers

2k

views

Prove two commutative linear transformations on a vector space over an algebraically closed field can be simultaneously triangularized

linear-algebra vector-spaces transformation linear-transformations

Dec 6 '14 at 23:55 Dimitar Ho 1,441

1

vote

1

answer

201

views

Show if $\mathfrak p$ is a minimal prime ideal of $R$ then $\mathfrak pR_\mathfrak p$ is the only prime ideal of $R_\mathfrak p$

abstract-algebra ring-theory ideals localization

Jul 3 at 19:05 Bernard 162k

1

vote

1

answer

109

views

$λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$, give necessary and suﬃcient conditions on $f,g$ for $λ ⊥ ρ$ and $λ << ρ$

real-analysis analysis measure-theory lebesgue-measure singular-measures

Feb 11 '16 at 5:53 user138017 991

1

vote

2

answers

573

views

Show that the dual space of a subspace of $V$ can be identified with $V$

linear-algebra vector-spaces duality-theorems

Sep 25 '15 at 10:59 Amitai Yuval 17.5k

1

vote

1

answer

94

views

Find the dimension of a subspace

linear-algebra polynomials vector-spaces

Sep 19 '15 at 7:21 Deliasaghi 690

1

vote

1

answer

87

views

If $I$ proper ideal of $R$, $S$ ring extension of $R$, and $u$ a unit in $S$, then $IR[u] \ne R[u]$ or $IR[u^{−1}] \ne R[u^{-1}]$

abstract-algebra ring-theory commutative-algebra extension-field

Apr 9 '15 at 1:27 Bill Dubuque 244k

1

vote

1

answer

51

views

Find the unique lifting for $f:S^1 \to S^1$ given by $f(z)=z^2$, where $z=x+iy, x^2+y^2=1$

general-topology universal-property

Mar 18 '15 at 6:16 user138017 991

1

vote

2

answers

290

views

If $X$ and $Y$ are g-equivariant homeomorphic then $X/G$ and $Y/G$ are homeomorphic

general-topology group-theory group-actions quotient-spaces

Mar 3 '15 at 2:23 Brian M. Scott 550k

1

vote

2

answers

135

views

Topological group closed path

group-theory algebraic-topology homotopy-theory fundamental-groups

Feb 17 '15 at 15:02 Ronnie Brown 13.7k

1

vote

2

answers

168

views

If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, then $QR_S$ is a radical ideal of $R_S$

abstract-algebra ring-theory ideals localization

Feb 16 '15 at 5:40 Rachmaninoff 2,430

1

vote

1

answer

136

views

Prove that for ideals I and J of a commutative ring, $√(I∩J)=√(IJ)$

abstract-algebra ideals

Feb 9 '15 at 9:01 user26857 42.7k

1

vote

1

answer

1k

views

Prove the rank of the direct sum of two linear transformations (on finite-dimensional vector spaces) is the sum of their ranks.

linear-algebra vector-spaces transformation

Dec 6 '14 at 5:04 user138017 991

1

vote

1

answer

135

views

Under what conditions is the operator $Ax = [x, y]x'$ a projection?

linear-algebra vector-spaces transformation

Dec 5 '14 at 8:16 Marc van Leeuwen 99.7k

0

votes

1

answer

765

views

show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent.

abstract-algebra ring-theory ideals localization

Jan 26 '19 at 8:57 Eric Wofsey 259k

0

votes

1

answer

36

views

Compare $f(x_1)+f(x_2)$ with $f(x_1+x_2)$ if $xf'(x)>f(x)$ holds for all $x\in (-\infty,0)$

calculus

Nov 7 '18 at 2:46 Riemann 5,421

0

votes

1

answer

196

views

Show that $\mathbb{R}^2/∼$ is homeomorphic to $\mathbb{R}$

general-topology equivalence-relations quotient-spaces

Jul 27 '18 at 3:26 mathworker21 29.2k

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53 Questions

Suppose $e^A = A$, prove that $A$ is diagonalizable

Is there always an intermediate field between non-prime field extension?

if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian [duplicate]

If $A, B$ are positive definite and $\|A+B\| = \|A\|+\|B\|$, they have a common eigenvector

Show the projective space $RP^2$ minus a point is homotopy equivalent to the unit circle $S^1$

A number related to the roots of a quartic polynomial is a root of a cubic polynomial

Show an ideal is a finitely generated projective module via a split exact sequence

orthogonal matrix with negative determinant

$A,B$ are orthogonal projections and $\|Ax\|^2+\|Bx\|^2=\|x\|^2$ show $A+B=I$

Exercise 3.40 from Folland, Real Analysis

How to use Seifert–van Kampen to find $\pi_1( RP^n ∨ RP^n)$

Show $(⊕M_α)/I(⊕M_α)\simeq ⊕(M_α/IM_α)$ for ${M_α}$ $R$-modules, and $I$ is an ideal of the commutative ring $R$

homotopic closed paths in topological group

The finitely generated-ness of ideals $I +rR$ and $I:r$ imply $I$ is a finitely generated ideal [closed]

Prove two commutative linear transformations on a vector space over an algebraically closed field can be simultaneously triangularized

Show if $\mathfrak p$ is a minimal prime ideal of $R$ then $\mathfrak pR_\mathfrak p$ is the only prime ideal of $R_\mathfrak p$

$λ = f dµ$ and $ρ = g dµ$ for Lebesgue measure $µ$, give necessary and suﬃcient conditions on $f,g$ for $λ ⊥ ρ$ and $λ << ρ$

Show that the dual space of a subspace of $V$ can be identified with $V$

Find the dimension of a subspace

If $I$ proper ideal of $R$, $S$ ring extension of $R$, and $u$ a unit in $S$, then $IR[u] \ne R[u]$ or $IR[u^{−1}] \ne R[u^{-1}]$

Find the unique lifting for $f:S^1 \to S^1$ given by $f(z)=z^2$, where $z=x+iy, x^2+y^2=1$

If $X$ and $Y$ are g-equivariant homeomorphic then $X/G$ and $Y/G$ are homeomorphic

Topological group closed path

If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, then $QR_S$ is a radical ideal of $R_S$

Prove that for ideals I and J of a commutative ring, $√(I∩J)=√(IJ)$

Prove the rank of the direct sum of two linear transformations (on finite-dimensional vector spaces) is the sum of their ranks.

Under what conditions is the operator $Ax = [x, y]x'$ a projection?

show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent.

Compare $f(x_1)+f(x_2)$ with $f(x_1+x_2)$ if $xf'(x)>f(x)$ holds for all $x\in (-\infty,0)$

Show that $\mathbb{R}^2/∼$ is homeomorphic to $\mathbb{R}$