Von Neumann ring

Let $(\mathcal{R}, +, \cdot)$ be a ring without necessary a unitary element but it has at least two elements. If $\mathcal{R}$ furthermore satisfies the property: "for every element $a \in \mathcal{R}$ that is not zero ($a \neq 0$) there exists a unique element $b \in \mathcal{R}$ such that $aba=a$ , then prove that:

1. $\mathcal{R}$ does not have zero divisors.
2. $bab=b$.
3. $\mathcal{R}$ has a unitary element.
4. $\mathcal{R}$ is a divisor ring.

Solution

An exercise on a ring

Let $\mathcal{R}$ be a ring and we suppose that $a, b \in \mathcal{R}$ be two elements of $\mathcal{R}$ such that $ab=1$. Prove that:

1. $a^n b^n =1$ forall $n \geq 1$.
2. $(1-ba)b^n =0$ forall $n \geq 1$.
3. the elements $ba$ and $1-ba$ are idempotent and orthogonal.
4. if $c \in \mathcal{R}$ and $ca=1$ , then the element $a$ is invertible and it holds that:
   
   $$a^{-1}=b=c$$

Solution

Some examples on rings

Give an example on the following:

1.  A commutative ring that is not a field but its characteristic is a prime number.
2. An example of a ring that has infinite elements but a finite characteristic.
3. An integral domain that has a finite characteristic.

Solution

Finite cyclic group and generator

Let $G=\langle a \rangle$ be a finite cyclic group of order $n$ with generator $a$. Prove that $a^k$ generates $G$ if and only if $\gcd(n, k) =1$.

Solution

Abelian Group of order p^2

Prove that every group of order $p^2$, where $p$ is prime, is an abelian group.

Solution

Algebraic equation

Solve the equation

$$x+\sqrt{2+\sqrt{2+\sqrt{x}}}=2$$

Solution

On ring theory

Let $f:R \rightarrow S$ be a ring homomorphism that is onto $S$. If the ring $(R, +, \cdot)$ is commutative then prove that the ring $(S, +, \cdot)$ is commutative. In continuity , give an example that the converse is not true.

Solution

Binomial sum invoking roots of unity

Evalute (in a closed form) the sum:

$$\sum_{k=0}^{n}\binom{3n}{3k}$$

Solution

Ideal and maximal ideal

Let $\left(C(\left[0,1\right]),+,\cdot\right)$ be the commutative ring with unity, of all continuous functions $f:\left[0,1\right]\rightarrow \mathbb{R}$. For each $X\in\mathbb{P}(\left[0,1\right])-\left\{\varnothing\right\}$, define the set
$$I_{X}=\left\{f\in C(\left[0,1\right]): f(x)=0, \,\forall\,x\in X\right\}$$

 a) Prove that the set $I_{X}$is an ideal of the ring $\left(C(\left[0,1\right]),+,\cdot\right)$ for each $X\in\mathbb{P}(\left[0,1\right])-\left\{\varnothing\right\}$.

 b) If  $X=\left\{x_0\right\}\subseteq \left[0,1\right]$ for some $x_0\in\left[0,1\right]$, then prove that the ideal $I_{X}$ is maximal.

Algebra and Topology

Let $(X, \mathbb{T})$ be a topological space. Consider the commutative ring with unity $\left(C(X,\mathbb{R}),+,\cdot\right)$ of all continuous functions $f: X \rightarrow \mathbb{R}$ . Prove that the topological space $(X, \mathbb{T})$ is connected if-f the ring $\left(C(X,\mathbb{R}),+,\cdot\right)$ is connected.

Solution