Intereting Posts

Set of points of $[0,1)$ that have a unique binary expansion
What is the the $n$ times iteration of $f(x)=\frac{x}{\sqrt{1+x^2}}$?
Proof: in $\mathbb{R}$, $((0,1),|\cdot|)$ is not compact.
How many ways are there to express a number as the product of groups of three of its factors?
PDF of a sum of exponential random variables
Convergence in topologies
Why does not $\int_{-\infty}^\infty x\,\mathrm{d}x$ converge?
How do you integrate $\int_0^\infty \exp(it^k)\,\mathrm dt$ for $k \in \Bbb N$?
Computing Residues
Product rule with stochastic differentials
Why do we represent complex numbers as the sum of real and imaginary parts?
Markov chain: join states in Transition Matrix
Meaning of “kernel”
A basic question on expectation of distribution composed random variables
Show that the function $g(x) = x^2 \sin(\frac{1}{x}) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$

Based on the comments of rschwieb’s answer in this question asked recently: Can we contruct a basis in a finitely generated module.

If $M=\langle e_1,\ldots,e_n\rangle$ is a finitely generated torsion free $R$-module. I’m trying to construct a free submodule $F$, i.e, isomorphic to $R^s$ for some $s$, finding a subset $S=\{e_1,\ldots,e_s\}$ such that $S$ is a maximal independent subset of $M$, then $S$ generates this free submodule $F$ of $M$ with basis $S$.

I’m asking that because I didn’t understand why Peter Clark in his commutative algebra pdf wrote this:

- Question about radical of a module.
- Applications of the Isomorphism theorems
- Intermediate fields Separable, Algebraic, or Splitting
- Is the localization of a PID a PID?
- Splitting field of $x^{n}-1$ over $\mathbb{Q}$
- Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$

Thanks in advance

- $ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
- An ideal is homogeneous if and only if it can be generated by homogeneous elements
- For which $k$ do the $k$th powers of the roots of a polynomial give a basis for a number field?
- The group structure of elliptic curve over $\mathbb F_p$
- Does the triangle inequality follow from the rest of the properties of a subfield-valued absolute value?
- An automorphism of the field of $p$-adic numbers
- Question on group homomorphisms involving the standard Z-basis
- Is Belnap's four valued-logic a boolean algebra?
- good books on Abstract Algebra and Cryptography for self-study
- Subgroups containing kernel of group morphism to an abelian group are normal.

Let’s call $S$ the L.I. subsets of $\{x_1,\ldots,x_n\}$. Since $M$ is a torsion free, for $x_1\in M,r\in R$, we have $rx_1=0$ iff $r=0$. Then $S$ is non-empty.

Since $\{x_1,\ldots,x_n\}$ is finite, then it has a maximal element $\{x_1,\ldots, x_s\}$, $1\le s\le n$

- What's a BETTER way to see the Gauss's composition law for binary quadratic forms?
- Prove $\lim_{x\rightarrow 0}\cos (x)=1$ with the epsilon-delta definition of limits
- Prove that if $A$ is transitive, then $\mathcal{P}(A)$ is transitive too.
- How does scaling $\Pr(B|A)$ with $\Pr(A)$ mean multiplying them together?
- Gathering books on Lorentzian Geometry
- Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
- Proving $nk = kn$
- What sequence does $\frac{1}{1-s-s^4}$ generate?
- Formal System and Formal Logical System
- On the spectrum of the sum of two commuting elements in a Banach algebra
- Prove that $(x+y) \text{ mod } n = ((x \text{ mod } n)+(y \text{ mod } n)) \text{ mod } n$
- Hahn-Banach theorem: 2 versions
- Number of ways to put N indistinct objects into M indistinct boxes
- Gaussian Integers and Quotient Rings
- How to solve the limit? $ \lim_{x \to +\infty} \frac{(\int^x_0e^{x^2}dx)^2}{\int_0^xe^{2x^2}dx}$