Domača naloga KomAlg.2

1. Naj bo R kolobar in M modul nad R. Dokaži: če obstaja u\in M, da je M=R\cdot u, potem je M\cong R/{\rm Ann\,}(u).

2. Naj bo M končno generiran R-modul in f:M\to M homomorfizem modulov. Pokaži: če je f surjektiven, je f izomorfizem.

3. Podan je komutativen diagram R-modulskih homomorfizmov \begin{matrix} A & \stackrel{f}{\longrightarrow} & B & \stackrel{g}{\longrightarrow} & C \\ \downarrow\alpha & & \downarrow\beta & & \downarrow\gamma\\ A' & \stackrel{f'}{\longrightarrow} & B' & \stackrel{g'}{\longrightarrow} & C'\end{matrix}. Predpostavimo tudi, da sta obe vrstici eksaktni. Dokaži:

  • \beta({\rm im\,}\beta \bigcap{\rm im\,}f')=\beta({\rm ker\,}(\gamma\circ g));
  • {\rm im\,}(f'\circ \alpha)=\beta( {\rm ker\,}\beta+{\rm ker\,}g).

4. Izračunaj naslednje tenzorske produkte:

  • \mathbb Q\otimes_{\mathbb Z}\mathbb Q;
  • \mathbb Q\otimes_{\mathbb Z}\mathbb Z_n;
  • \mathbb Z[X]\otimes_{\mathbb Z}\mathbb Z[X];
  • M_n(\mathbb C)\otimes_{\mathbb C}M_m(\mathbb C).

5. Naj bo \mathfrak a\vartriangleleft R in M modul nad R. Pokaži, da je (R/\mathfrak a)\otimes_R M \cong M/(\mathfrak aM).

rok oddaje: ponedeljek, 4. januar 2010 ob 16:30.

This entry was posted in Komutativna algebra (FNM), Pedagoško delo, Slovenščina. Bookmark the permalink.

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