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11

A proof that every projective module over a PID is free occurs in $\S$ 3.9 of my commutative algebra notes. As Qiaochu Yuan mentions, infinitely generated projective modules long to be free. A generalization of Kaplansky's result is a 1963 theorem of H. Bass: let $R$ be a connected (i.e., without nontrivial idempotents) Noetherian ring. Then every ...


9

Let $M, N$ be $R-$modules. Then the following holds. If $M$ and $N$ is flat, then so is $M\otimes_{R}N$: see related question here. If $M$ and $N$ are projective, then so is $M\otimes_{R} N$. Indeed, writing $M\oplus M'=F,\ N\oplus N'=F'$, for free $R-$modules $F,\ F'$, one has that $$ F'':=F\otimes_{R}F' $$ is free (tensor product of free modules) and ...


8

I'm denoting your ring with $R$ and your ideal with $I$. We'll just need that $R$ is an integral domain, and that $I$ is a nontrivial ideal. If $R/I$ is a projective $R$ module, then the following exact sequence splits $$ 0\rightarrow I\rightarrow R\rightarrow R/I\rightarrow 0 $$ But integral domains have no proper direct summands, so $I$ would have to be ...


8

As indicated in the comments, you should saturate $Q$ in $P$ first, i.e. replace it by the preimage of the torsion in $P/Q,$ so that (after changing $Q$ in this way) we get that $P/Q$ is torsion-free. The basic fact you need is that the saturation is again f.g., but this will follows from the fact that the torsion in $Q/P$ is f.g., being a submodule of the ...


7

This failure of freeness is a non-trivial result. One way to prove it is to begin with a lemma: If $F$ is a free abelian group and $C$ is a countable subgroup, then the quotient $F/C$ is the direct sum of a countable group and a free group. (I'm omitting "abelian" because I'm lazy and all groups here will be abelian.) [Proof of lemma: Fix a basis $B$ for ...


7

Since $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \cong \mathbb{Q}$ is a field, every module over it is projective.


6

The truth is (to me) quite surprising: Kaplansky showed that an infinitely generated projective module over any Dedekind domain $D$ is free! (The corresponding statement for finitely generated projective modules is equivalent to $D$ having trivial class group.) This is referenced, for example, here.


6

Here are some examples: Localizations or quotients of $R$. More generally any epimorphism of commutative rings with domain $R$. Free $R$-modules of infinite rank. $R \oplus \bigoplus_{i \in I} N$ for any $R$-module $N$ with $N \otimes N = 0$. The finitely generated examples can be classified: Claim: When $M$ is a finitely generated $R$-module with $M ...


6

A projective module over a domain has no nonzero torsion element, since it is a submodule of a free module. But every element of your module is a torsion element: it is killed by $x$.


6

As Steve D said, you can use the fact that projective modules are always flat. Consider the map $\mathbb{Q}[x,y]\to \mathbb{Q}[x,y]$ defined by multiplying $x$. This is an injective $\mathbb{Q}[x,y]$-module map, while tensoring $\mathbb{Q}$ will give an injective map, but it is NOT. So $\mathbb{Q}$ is not flat as $\mathbb{Q}[x,y]$-module. However, ...


6

Let $R$ be any commutative ring whose projective modules are all free, and let $e\notin \{0,1\}$ be an idempotent of $R$. Then $eR$ and $(1-e)R$ are both projective, hence free of some rank 1 or more, and $eR\oplus(1-e)R=R$, so that we have $R^n\cong R$ as $R$ module for some natural number $n\geq 2$. This is absurd since commutative rings have IBN. This ...


6

I'll consider the interval $[0,2\pi]$ for notational simplicity. Consider the matrix $$ A = \left( \begin{array}{cc} \sin ^2\tfrac{\theta }{2} & - \sin \tfrac{\theta }{2} \cos \tfrac{\theta }{2} \\ -\sin \tfrac{\theta }{2}\cos \tfrac{\theta }{2} & \cos ^2\tfrac{\theta }{2} \end{array} \right), $$ which defines an $R$-linear ...


6

Here is the answer for finitely generated modules of rank one. Recall that the isomorphism classes of these modules form a group, the Picard group $Pic(R)$, with tensor product as multiplication. Theorem (Traverso, Swan) For a commutative ring $R$ the following are equivalent: a) The reduced ring $R_{red}=R/Nil(R)$ is semi-normal b) The ...


5

Thinking of a f.g. projective module as a vector bundle, it seems very likely the answer is no: consider the trivial bundle of rank 2 and two "twisted" subbundles of rank 1 whose intersection is 0-dimensional everywhere except over a closed subset with non-empty interior – so not a vector bundle, in particular. Let's see where this thinking leads. Let $R$ ...


5

I don't know of any other meaning of a projective ideal other than the one suggested by Boris Novikov, i.e. an ideal of a ring $R$ that is also projective as an $R$-module. I want to emphasize that such an ideal $I$ need NOT be a direct summand of $R$ (Boris never implied that condition to be necessary - only sufficient!) as well as give more examples. The ...


5

This has been answered on MathOverflow here.


5

Please see $\S 3.5.4$ -- "Projective verus free" -- in my commutative algebra notes. In particular, Proposition 27 and the exercise follow it step you through showing that the ideal $\langle 3, 1+ \sqrt{-5} \rangle$ is projective but not free. The same techniques apply to $I = \langle 2, 1+ \sqrt{-5} \rangle$. (In fact, I view it as a happy accident ...


5

For the case $R$ is a local ring it's a corollary of Nakayama's lemma. As the notation in the above link, suppose $M$ is a finite generated projective module over $R$, then, first pick a minimal number of generators, i.e., $M=Rm_1+\cdots +Rm_k$, and $k$ is the minimal number with this property, so we get a decomposition $$R^k=M\oplus N,$$ then, we are ...


5

The statement "every element in P can be written as a finite linear combination of some elements of P.", where "some" means a finite set, just says that the module is finitely generated. This has nothing to do with being projective. Take for instance the $\mathbb Z$-module $\mathbb Z/2$. Here every element can be written as a multiple of $[1]$. So $\mathbb ...


5

The statement you're linking to is: A module $P$ is projective if and only if there is a family $\{x_{i}\}_{i \in I} \subset P$ and morphisms $f_{i}: P \to R$ such that for each $x \in P$ we have $x = \sum_{i \in I} f_{i}(x) x_{i}$. The last statement says three things: In order for the sum to make sense we must have that for all $x$ the set ...


4

As Andrew pointed out, every finitely-generated projective module over a local ring is free (in fact, the hypothesis of being finitely-generated can be dropped - this is a theorem of Kaplansky). Hence, it remains to show that there exists a non-free module. But we have the following characterisation: A commutative ring $R$ is a field if and only if every ...


4

A few remarks, to be expanded below: (1) first is that the proof that $M = \prod_{i=1}^\infty\mathbb{Z}$ is not free is elementary, and (2) second is that it might be hard to find simpler examples, at least if "simple" refers to how simple the ring is itself. (1) In fact, here's a proof that I learned from Kaplansky's book "Infinite Abelian Groups": Assume ...


4

Sure. Projective modules $P$ have the property (and actually this is an equivalent characterization) that every epimorphism $F \to P$ splits. Now choose a finite generating system of $P$, this lets you choose $F$ finitely generated free. Of course every direct summand of $F$ is a quotient of $F$ and therefore also finitely generated.


4

Finitely generated and torsion free implies projective, but infinitely generated torsion free modules need not be projective. (E.g. the fraction field of $A$ is not projective as an $A$-module; think about $\mathbb Q$ over $\mathbb Z$.) This is why you are having trouble proving it; you have to use finitely generated in the argument. (More generally, over ...


4

Let $R=K^{\mathbb N}$ be a countable direct product of copies of a field $K$. (For simplicity we may suppose $K=\mathbb Z/2\mathbb Z$.) This ring is self-injective, and its global dimension is $2$. But $I=K^{(\mathbb N)}$, a countable direct sum of copies of $K$, is an ideal of $R$ which is not a direct summand of $R$, and therefore $R/I$ isn't projective. ...


4

Consider the surjection $R^n \to R$ described by the OP, namely $$(x_1,\ldots,x_n) \mapsto \sum_i r_i x_i.$$ This is a surjection $R^n \to R$. Since $R$ is a free as a module over iself, we may split this surjection. (Concretely, write $1 = \sum_i a_i r_i,$ and define a splitting via $r \mapsto (ra_1,\ldots,r a_n).$) Thus $R^n \cong M \oplus R,$ and so ...


4

Hint: $P\oplus P'=R^{(\Lambda)}$ (a free module) for some set $\Lambda$ and some module $P'$.



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