Intereting Posts

An application of partitions of unity: integrating over open sets.
Please prove: $ \lim_{n\to \infty}\sqrt{\frac{1}{n!}} = 0 $
Can every set be expressed as the union of a chain of sets of lesser cardinality?
Upper bound on cardinality of a field
Solve three equations with three unknowns
Isometry from Banach Space to a Normed linear space maps
If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?
What is the support of a localised module?
Why this polynomial is irreducible?
Find a Recurrence Relation
Help on showing a function is Riemann integrable
Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$
What is the cross product in spherical coordinates?
Evaluating $\lim_{n\to\infty} \frac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ using integral

Let $B$ be a commutative ring.

Let $A$ be a subring of $B$.

If $M$ and $N$ are $\mathbb{Z}$-submodules of $B$,

we denote by $MN$ the submodule of $B$ generated by the subset $\{ab\mid a \in M, b\in N\}$.

If $M$ and $N$ are $A$-submodules of $B$, $MN$ is clearly an $A$-submodule of $B$.

If $M$ and $N$ are $A$-submodules of $B$, we denote by $(N : M)$ the set $\{x \in B\mid xM \subset N\}$.

$(N : M)$ is clearly an $A$-submodule of $B$.

Is the following proposition correct?

If yes, how do we prove it?

- If $A$ is a Dedekind domain and $I \subset A$ a non-zero ideal, then every ideal of $A/I$ is principal.
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- Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition
- $A/ I \otimes_A A/J \cong A/(I+J)$
- $M\oplus A \cong A\oplus A$ implies $M\cong A$?
- isomorphic ideals and projective dimensions of quotients

**Proposition.**

Let $M$ be an $A$-submodule of $B$.

Suppose there exists an $A$-submodule $N$ of $B$ such that $MN = A$.

Then $N = (A : M)$.

- Computing with ideals: over $K$ or over $\mathbb{Q}\subseteq K$? does it matter?
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- Explicit examples of infinitely many irreducible polynomials in k
- Support of a module with extended scalars
- When does locally irreducible imply irreducible?
- Sheafyness and relative chinese remainder theorem

Yes, this is correct, and easy to prove.

$MN=A$ implies $N \subseteq (A:M)$ and $(A:M) = (A:M)MN \subseteq AN \subseteq N$, hence $(A:M)=N$.

- Integers of the form $a^2+b^2+c^3+d^3$
- Is $2^{16} = 65536$ the only power of $2$ that has no digit which is a power of $2$ in base-$10$?
- Complex numbers modulo integers
- Evaluate $\displaystyle \sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x.$
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- Proof Verification : Prove -(-a)=a using only ordered field axioms
- Any good approximation for this integral?
- Understanding a proof in MacLane-Moerdijk's “Sheaves in Geometry and Logic”
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- Why is the Tangent bundle orientable?
- What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$?
- Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.