Projective Modules

Let \(R\) be a commutative ring. Let \(P\) and \(Q\) be two projective \(R\)-modules. Prove that \(P\otimes _R Q\) is projective.

Let \(R:=\mathbb {Q}[x]/(x^3-1)\). Give an example of a finitely generated projective \(R\)-module that is not free.

Let \(R\) be a commutative ring. Let \(P\) and \(Q\) be two projective \(R\)-modules. Prove that \(P\otimes _R Q\) is projective.

Prove that the additive group of \(\mathbb {Q}\) is not a projective \(\mathbb {Z}\)-module.

Consider the ideal \(I=(2,1+\sqrt {-5})\) in the ring \(R=\mathbb {Z}[\sqrt {-5}]\). Is \(I\) a prime ideal? Is \(I\) a projective \(R\)-module?

Prove that the \(\mathbb {Z}\)-module \(\mathbb {Q}\) is not projective.

In the category of \(\mathbb {Z}\)-modules:

1. Is \(\mathbb {Z}\) injective?
2. Is \(\mathbb {Z}/8\mathbb {Z}\) projective?

Prove that the \(\mathbb {Z}\)-module \(\mathbb {Q}\) is not projective.

In the category of \(\mathbb {Z}\)-modules, is the module \(\mathbb {Q}/\mathbb {Z}\) (a) projective? (b) injective? (c) flat?

Justify your answer.

Let \(R\) be a commutative ring with 1 and \[ 0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0,\qquad 0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0 \] be short exact sequences of \(R\)-modules where \(P\) and \(P'\) are projective \(R\)-modules. Show that there is an isomorphism of \(R\)-modules \(K\oplus P'\cong K'\oplus P\).

Let \(R\) be a ring with identity. Suppose \[ 0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0 \] and \[ 0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0 \] are two short exact sequences of \(R\)-modules. Suppose further that \(P\) and \(P'\) are projective. Prove that \(K'\oplus P\cong K\oplus P'\) as \(R\)-modules.

Let \(P_\bullet \) be a complex of projective modules with \(P_n=0\) for \(n<0\). Let \(E_\bullet \) be an exact complex. Show that any map of complexes from \(P_\bullet \) to \(E_\bullet \) is homotopic to zero.

Show that if \[ 0\to K\to P\to M\to 0 \] and \[ 0\to K'\to P'\to M\to 0 \] are short exact sequences of \(R\)-modules, where \(P\) and \(P'\) are projective, then \[ K\oplus P'\cong K'\oplus P. \]