Intereting Posts

Eigenvalues of symmetric matrix with skew-symmetric matrix perturbation
$ 0 < a < b\,\Rightarrow\, b\bmod p\, <\, a\bmod p\ $ for some prime $p$
Probability: 10th ball is blue
A consequence of the Mean Value Theorem for Integrals
Factorial number of digits
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Proving that $\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$
A confusion about Axiom of Choice and existence of maximal ideals.
Predicates and Quantifiers?
Example about the Reduced cost in the Big-M method?
Problem evaluating $ \int_{0}^{\pi/2}\frac{x}{\tan x}dx $
Z coordinates of 3rd point (vertex) of a right triangle given all data in 3D
Counting $k$-ary labelled trees
Making the water gallon brainteaser rigorous
Is there a general formula for estimating the step size h in numerical differentiation formulas?

I need two counterexamples.

First, a direct sum of $R$-modules is projective iff each one is projective.

But I need an example to show that, “an arbitrary direct product of projective modules need not be a projective module.”

If I let $R= \mathbb Z$ then $\mathbb Z$ is a projective $R$-module, but the direct product $\mathbb Z \times \mathbb Z \times \cdots$ is not free, hence it is not a projective module. We have a theorem which says that every free module over a ring $R$ is projective. Am I correct?

- Which of the following is not true?
- If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
- Minimal polynomial over an extension field divides the minimal polynomial over the base field
- Classification of indecomposable modules over a given ring
- why is a polycyclic group that is residually finite p-group nilpotent?
- How should I show that the Lie algebra so(6) of SO(6) is isomorphic to the Lie algebra su(4) of SU(4)?

Second, a direct product of $R$-modules is injective iff each one is injective

but I need an example to show that the direct sum of injective modules need not be injective.

- Does $g = x^m - 1 \mid x^{mk} - 1$ for any $k \in \mathbb{N}$?
- Are there real world applications of finite group theory?
- Support of a module with extended scalars
- Exponential fields as structures with three binary operations.
- Is $A \times B$ the same as $A \oplus B$?
- Tarski Monster group with prime $3$ or $5$
- Norm of powers of a maximal ideal (in residually finite rings)
- Irreducible and prime elements
- $M_1$ and $M_2$ are subgroups and $M_1/N=M_2/N$. Is $M_1\cong M_2$?
- If $M/N$ and $N$ are noetherian $R$-modules then so is $M$

As for the first question: yes, $P = \prod_{i=1}^{\infty} \mathbb{Z}$ is a direct product of free $\mathbb{Z}$-modules which is not free. Since $\mathbb{Z}$ is a PID, $P$ is also not projective. The proof that $P$ is not free is nontrivial, but I believe it has already been given either here or on MathOverflow.

As for the second question: the **Bass-Papp Theorem** asserts that a commutative ring $R$ is Noetherian iff every direct sum of injective $R$-modules is injective. Thus every non-Noetherian ring carries a counterexample. The proof of the result — given for instance in $\S 8.9$ of these notes — is reasonably constructive: if

$I_1 \subsetneq I_2 \subsetneq \ldots \subsetneq I_n \subsetneq \ldots$

is an infinite properly ascending chain of ideals of $R$, then for all $n$ let $E_n =

E(R/I_n)$ be the **injective envelope** (see $\S 3.6.5$ of loc. cit.) of $R/I_n$, and let $E = \bigoplus_{n=1}^{\infty} E_n$. Then $E$ is a direct sum of injective modules and (an argument given in the notes shows) that $E$ is not itself injective.

- Proof of the second Bianchi identity
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- Show that these two numbers have the same number of digits
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- Semi-Norms and the Definition of the Weak Topology
- half sine and half cosine quaternions
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- Surjectivity implies injectivity
- Cofinality of cardinals
- generalized inverse of a matrix and convergence for singular matrix