I am curious about "projective" and "injective" modules. A projective module $P$ is by definition one for which whenever one have morphisms $\eta:P\to B$ and $\mu:A\to B$ a surjection, we can "factor" $\eta$ with a third $\varphi:P\to A$, as in $\eta=\mu\varphi$, correct?
Many (co)homology theories are defined with resolutions, as Karl says, but there are a lot of nice exact sequences associates to homology in various ways the you actually use to compute the homology.
@FernandoMartin (The categories of modules over) Integral domains that aren't fields are ruled out, since every ideal would be a direct summand of $R$.
Hm, not sure what modules over $k\times k$, where $k$, is a field are like. That'd be the most likely nontrivial possibility though.
Whenever I try to compile a file using PDFLaTeX, I get "Error, file not found."
I have no idea what is causing this problem. Could it be I am opening a code that uses a package I haven't installed? Usually TeXWorks fixes this automatically by downloading such package. What the issue here?
@KarlKronenfeld Suppose I have a ring $A$ and an abelian group $G$. Then $A[G]$ is an $A$-module in the most obvious way, right? I mean, if we take the additive group of that ring.
For every group (monoid) morphism $\varphi:G\to G'$ there is a unique ring morphism $\bar\varphi:A[G]\to A[G']$ extending $\varphi$ (in the isomorphic copy of $G$ inside $A[G]$), which is the identity on $A$. @KarlKronenfeld
@eXtremiity Do you know about the trivial and discrete topology?
OK, you want every map to be continuous. Think about what that would mean in terms of delta-epsilon. Maybe fiddle with it on paper, play with some metrics you know.
@Pedro: Sorry if you already mentioned this stuff the other day but I suck at understanding things without writing them down and thinking about them for a while
Suppose $A$ is an $n\times m$ matrix with coefficients in $\{0,1\}$ such that every row and every column has an odd number of $1$s. Then $m,n$ have the same parity.