Intereting Posts

Inverse of diagonally dominant matrix with equal off-diagonal entries
Local ring on generic fiber
Is it possible to create a completely random integer between 1 and 13 using standard dice in a D&D dice kit?
Continuity of Parametric Integral
Maximum of the sum of cube
$\int_{-\infty}^\infty e^{ikx}dx$ equals what?
A question about proving that there is no greatest common divisor
Prove that there is no smallest positive real number
On critical points of a large function
Regarding nowhere dense subsets and their measure.
Suppose that $f$ is a real valued function such that its second derivative is discontinuous.Can you give some example?
On the number divisors of a number
When does every group with order divisible by $n$ have a subgroup of order $n$?
For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$
Classsifying 1- and 2- dimensional Algebras, up to Isomorphism

In a module, we know what a minimal generating set is. But, is it always true that such a set exists? If the module is finitely generated, is it possible?

- every field of characteristic 0 has a discrete valuation ring?
- A commutative ring in which every prime ideal is 2-generated
- Does UFD imply noetherian?
- Nilradical strictly smaller than Jacobson radical.
- Contraction of non-zero prime ideals in the ring of algebraic integers
- Every element in a ring with finitely many ideals is either a unit or a zero divisor.
- What is the kernel of $K \to K$, defined by $T \mapsto x$?
- Is there an example of a non-noetherian one-dimensional UFD?
- Can you construct a field with 4 elements?
- Exercise 5.5.F. on Ravi Vakil's Notes related to associated points

Not every module has minimal generating sets. As another example on the same vein as Hurkyl’s, consider the Prüfer $p$-group as a $\mathbb{Z}$-module. A subset generates if and only if it contains elements of arbitrarily high order; but you can remove any finite subset of such a set (you can even remove infinite subsets) and still have a set with that property. Thus, no generating set is minimal: they all contains as proper subsets other generating sets.

It is also not true in general that two minimal generating sets, if they exist, will have the same size: $\mathbb{Z}$ has a minimal generating set (over itself) given by the single element $1$, but it also has a minimal generating set with two elements, $\{2,3\}$. And one with three elements: $\{6, 10, 15\}$. In fact, there is a minimal generating set with any finite number of elements.

Not every module has a minimal generating set. The $\mathbb{Z}$-module $\mathbb{Q}$, for example.

If a module is finitely generated, then the existence of a minimal generating set is easy to show: take any finite generating set and keep removing elements until you can’t anymore.

If M is a module over a ring (not necessarily commutative) which is **not** a finitely-generated module then every two minimal generating sets of M have the same cardinality (provided that at least a minimal generating set of M exists). This assertion (as stated in the above answers) in the finitely-generated case is not necessarily true. But, if the underlying ring is **commutative** we have then the following result:

**Every two bases of a free module have the same cardinality.**

- Injective function and ultrafilters
- Bound the Number of Acute-angled Triangles
- Subsets with small interaction.
- Probability returning to initial state
- There is no norm in $C^\infty ()$, which makes it a Banach space.
- $x^5 + y^2 = z^3$
- Proving for $n \ge 25$, $p_n > 3.75n$ where $p_n$ is the $n$th prime.
- Intuition behind quotient groups?
- A question in germs and multiplicity of zeroes.
- Proof of dilogarithm reflection formula $\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$
- Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ ?
- Finding 2 poisoned bottles of wine out of a 1000
- Are infinitesimals dangerous?
- Comparing the growth rates
- Much less than, what does that mean?