Intereting Posts

How to prove the ring of Laurent polynomials over a field is a principal ideal domain?
Derive Fourier transform of sinc function
Field extension obtained by adjoining a cubic root to the rationals.
$F$ is an equivalence of categories implies that $F$ is fully faithful and essentially surjective
How to evaluate the integral $\int e^{x^3}dx $
What are some strong algebraic number theory PhD programs?
How to make sense out of the $\epsilon-\delta$ definition of a limit?
International Mathematical Olympiad 1988 Problem 6: canonical solution?
Reading, Writing, and Proving Math: Cartesian Product
Homology of the Klein Bottle
Proving two equations involving the greatest common divisor
proof of l'Hôpital's rule
How to prove that $\sum_{r=0}^n\binom{n}{r}2^r=3^n$
Is it generally true that $\arcsin \theta + \arccos \theta = \frac{\pi}{2}$?
Is $\Delta_0=\Delta_1$ in arithmetical hierarchy?

Let $I=\langle a_1,\dots, a_s\rangle, J=\langle b_1,\dots, b_t\rangle$ be ideals of arbitrary commutative ring.

Then we know that

$I+J=\langle a_1,\dots, a_s, b_1,\dots, b_t\rangle, IJ=\langle\{a_ib_j \mid 1 \leq i \leq s, 1\leq j \leq t\}\rangle$.

Also $IJ\subseteq I\cap J \subseteq I+J$.

- Must an ideal contain the kernel for its image to be an ideal?
- An element of a group $G$ is not conjugate to its inverse if $\lvert G\rvert$ is odd
- Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
- Compact $n$-manifold has same integral cohomology as $S^n$?
- why is a polycyclic group that is residually finite p-group nilpotent?
- Finding a fixed subfield of $\mathbb{Q}(t)$

I wonder about the generators of $I\cap J$. Is it possible that know the generators? Or is it finitely generated?

- Is $\mathbb{Z} / \langle (x^2 + 1)^2 \rangle$ isomorphic to a familiar ring?
- How can I prove that every group of $N = 255$ elements is commutative?
- Quotient field of gaussian integers
- Spectrum of $\mathbb{Z}^\mathbb{N}$
- Proposition 5.21 in Atiyah-MacDonald
- What is the field of fractions of $\mathbb{Q}/(x^2+y^2)$?
- Constructing Idempotent Generator of Idempotent Ideal
- Prove that $G \cong\mathrm{Inn}(G)$ if and only if $Z(G)$ is trivial
- Show that any cyclic group of even order has exactly one element of order $2$
- System of linear equations having a real solution has also a rational solution.

The intersection of any two finitely generated ideals in an integral domain $R$ is also finitely generated if and only if $R$ is coherent. An example of GCD domain which is not coherent can be found in Example 4.4 of this paper. So,

there are GCD domains which have finitely generated ideals whose intersection is not finitely generated.

- If $x^2+ax+b=0$ has a rational root, show that it is in fact an integer
- Detecting that a fraction is a repeating decimal
- Relationship between the zeros of a vector field and the fixed points of its flow
- Prove that $\overline{A \cup B} = \overline A\cup \overline B$.
- Asymmetric random walk
- why do we use 'non-increasing' instead of decreasing?
- Prove that identity element is unique
- Finding the order of the automorphism group of the abelian group of order 8.
- Prove a group generated by two involutions is dihedral
- Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$
- All pairs shortest path in undirected and unweighted graphs
- Strictly increasing function with $f'(x) = f(f(x))$
- Matrix is conjugate to its own transpose
- Affine transformation
- How many cycles, $C_{4}$, does the graph $Q_{n}$ contain?