What is an example of a radical of sum of ideals not being equal to the sum of radicals?

Solution 1:

In $\mathbb Q[x,y]$ we have: $$\sqrt {(x)+(x-y^2)}=\sqrt{(x,y^2)}=(x,y)\neq \sqrt {(x)} +\sqrt {(x-y^2)}=(x)+(x-y^2)=(x,y^2)$$

[Optional remark: If you know the basic dictionary relating commutative rings to affine schemes, you will note that this is just an example of the phenomenon that the intersection of two reduced subschemes of some affine scheme needn't be reduced]

Infinite Series $\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{m!\:n!}{(m+n+2)!}$

If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$

Finding an equation for a circle given its center and a point through which it passes

Let $p$ be an odd prime. $p$ and $(1-\zeta_p)^{p-1}$ are associates in $\mathbb{Z}[\zeta_p]$.

The order of a conjugacy class is bounded by the index of the center

Why isn't an odd improper integral equal to zero

Wasserstein distance from a Dirac measure

Need a hint: show that a subset $E \subset \mathbb{R}$ with no limit points is at most countable.

To what fractional Sobolev spaces does the step function belong? (Sobolev-Slobodeckij norm of step function)

What does a well ordering of $\mathbb{R}$ look like? [duplicate]

There exists $C\neq0$ with $CA=BC$ iff $A$ and $B$ have a common eigenvalue

convergence in mean square implies convergence of variance