Blog

Taken from: [1, Exercise 9.42]. Exercise 9.42 Let \[0 \longrightarrow L \xrightarrow{\; f\;} M \xrightarrow{\; g\;} N \longrightarrow 0\] be a short exact sequence of modules and homomorphisms over the commutative Noetherian ring $R$. Prove that \[\operatorname{Ass}(L) \subseteq \operatorname{Ass}(M) \subseteq \operatorname{Ass}(L) \cup \op...

Taken from: [1, Exercise 5.43]. Exercise 5.43 Let $I$ be a decomposable ideal of the commutative ring $R$. Let \[I=Q_1 \cap \ldots \cap Q_n \quad \text { with } \sqrt{Q_i} =P_i \text { for } i=1, \ldots, n\] be a minimal primary decomposition of $I$. Let $\mathcal{P}$ be a non-empty subset of $\operatorname{ass}I$ with the property that when...

Taken from: [1, Exercise 4.36]. Exercise 4.36 Let $R$ be a commutative ring and let $X$ be an indeterminate; use the extension and contraction notation of 2.41 in conjunction with the natural ring homomorphism $f: R \rightarrow R[X]$. Let $Q$ and $I$ be ideals of $R$. Show that $Q$ is a primary ideal of $R$ if and only if $Q^e$ is a p...