Intereting Posts

Geometric interpretation of Euler's identity for homogeneous functions
Linear Algebra Complex Numbers
the concept of Mathematical Induction
Derivative of ${x^{x^2}}$
Proove that $2005$ devides $55555\dots$ with 800 5's
Taylor's Theorem with Peano's Form of Remainder
Tensor product of reduced $k$-algebras must be reduced?
Retracts are Submanifolds
Regularity of a quotient ring of the polynomial ring in three indeterminates
Determinant of a finite-dimensional matrix in terms of trace
Principal ideal and free module
Creating teams with exactly two men and one woman, where the order matters.
Show that $\mathbb{Z}$ has three ideal classes.
If $G/N$ and $H/N$ are isomorphic, does it imply $G$ and $H$ are isomorphic?
Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x).

All:

what are Goldbach conjectures for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, and other algebraic structures ?

In other word, for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, etc.

- Is a finite commutative ring with no zero-divisors always equal to the ideal generated by any of its nonzero elements
- Intersection maximal ideals of a polynomial ring
- Decomposition of polynomial into irreducible polynomials
- Ring of rational-coefficient power series defining entire functions
- find all the members of A4/K
- There are at least three mutually non-isomorphic rings with $4$ elements?

What does “even” element mean ?

What does “prime” element mean ?

Can every “even” element for those algebraic structures be partitioned into two “prime” elements ?

Has Goldbach Conjecture been proved for other algebraic structures ?

- The set of all nilpotent elements is an ideal
- Homomorphism and normal subgroups
- Possibilities for a group $G$ that acts faithfully on a set of objects with two orbits?
- Given that $a+b\sqrt{2} +c\sqrt{4} =0$, where $a,b,c$ are integers. Show $a=b=c=0$
- Which polynomials with binary coefficients evaluate only to 0 or 1 over an extension field?
- Transcendence degree of $K$
- Multiplication in Permutation Groups Written in Cyclic Notation
- Short Exact Sequences & Rank Nullity
- Center of Heisenberg group- Dummit and Foote, pg 54, 2.2
- On the order of elements of $GL(2,q)$?

Paul Pollack proved in this paper the following version of Goldbach for polynomial rings:

**Theorem (Pollack)**: let $R$ be a Noetherian ring with infinitely many maximal ideals. Then every polynomial in $R[x]$ of degree $n\ge 1$ can be written as the sum of two irreducible polynomials of degree $n$.

For $R=\mathbb{Z}$ this has been proved by David Hayes in $1965$.

- correspondence for universal subalgebras of $U/\vartheta$
- Probability that at least K cards will go into a bucket
- uniform convergence of sequence of functions $f_n(x)=x e^{-nx^2} , x\in{\mathbb{R}}$.
- Definition of local maxima, local minima
- Is there a way to determine how many digits a power of 2 will contain?
- Use Nakayama's Lemma to show that $I$ is principal, generated by an idempotent.
- Show that $\lim\limits_{n\rightarrow\infty} e^{-n}\sum\limits_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$
- Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
- Show that $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ is rational using the rational zeros theorem
- The sum of the first $n$ squares is a square: a system of two Pell-type-equations
- Endpoint-average inequality for a line segment in a normed space
- Number of combinations and permutations of letters
- Why is $f(x) = \sqrt{x}$ is continuous at $x=0$? Does $\lim_{x \to 0^{-}} \sqrt{x}$ exist?
- $\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$
- Non-associative commutative binary operation