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Let \[ 0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime \prime} \rightarrow 0 \] be an exact sequence of \(A\)-modules. Show that \( M \) is Noetherian iff \( M^{\prime} \) and \( M^{\prime \prime} \) are Noetherian. \( M \) is Artinian iff \( M^{\prime} \) and \( M^{\prime \prime} \) a...

Let \( \displaystyle X , Y \) be normed spaces and let \( \displaystyle B(X,Y) \) be the space of all bounded linear operators from \( \displaystyle X \) to \( \displaystyle Y \). Suppose that \( \displaystyle Y \) is a Banach space, let \( \displaystyle X_{0} \) be a dense subspace of \( \displayst...

Let \( I \) be a non-zero prime ideal of \( A \). By hypothesis, \[ \exists \ x \in A \ : I = (x) \]Let \( J \) be an ideal of \( A \) such that \( I \subsetneq J \subseteq A \). Again by hypothesis, \[ \exists \ y \in A \ : J = (y) \]Since \( x \in (x) \subset (y) \), \[ \exists \ z \in A \ : x = y...

Let \( A \) be an integral domain and let \( \mathbb{K} \) be its field of fractions. Show that the following are equivalent: \( A \) is a valuation ring of \( \mathbb{K} \). If \( \mathfrak{a},\mathfrak{b} \) are two ideals of \( A \), then either \( \mathfrak{a} \subseteq \mathfrak{b} \) or \( \ma...

My apologies, Vaggelis, for the mistake in the statement of the second part of the exercise! Define a map \[ \displaystyle \mu \ \colon A/I \times M \longrightarrow M \; , \; \left( a + I , m \right) \mapsto \left(a+I \right) \star m := am \] Then \( \star \) is well-defined, because \( I \subseteq ...

Show that

$$\int_{- \infty}^{+ \infty} {e}^{-2 \pi i x \xi} \frac{ \sin \pi \alpha }{ \cosh \pi x + \cos \pi \alpha } \mathrm{d} x = 2 \frac{ \sinh 2 \pi \alpha \xi }{ \sinh 2 \pi \xi }$$

where $\xi \in \mathbb{R}, \; \alpha \in (0, 1)$.

Let \( A \) be a ring (commutative and associative ring with unity), let \( \mathfrak{a} \) be an ideal of \( A \) and let \( M \) be an \( A \)-module. Show that \[ \displaystyle \left( A/ \mathfrak{a} \right) \otimes_{A} M \cong M / \mathfrak{a} M \]where \( \mathfrak{a} M \) is the submodule of \...

We follow the conventions of the previous posts regarding the ring \( \displaystyle A \). Let \( M \) be a finitely generated \( A \)-module and let \( \mathfrak{a} \) be an ideal of \( A \) contained in the Jacobson Radical \( \mathfrak{R} \) of \( A \). Show that if \( \mathfrak{a}M = M \), then \...

We follow the conventions of the previous posts regarding the ring \( \displaystyle A \). Let \( M \) be an \( A \)-module and let \( N \) and \( P \) be submodules of \( M \). Show that \[ \displaystyle ( N \, \colon P ) \; \; \colon = \left\{ a \in A \; \Big | \; aP \subseteq N \right\} \]is an id...

Show that the following properties of a morphism of schemes are not preserved under base change:

1. quasi-finite
2. injective
3. having reduced fibres
4. having integral fibres
5. having connected fibres

Is the property "surjective" preserved under base change?