Articles of abstract algebra

In a noetherian integral domain every non invertible element is a product of irreducible elements

I want to prove that in a noetherian ring $R$ which is also an integral domain, every non invertible element can be expressed as product of irreducible elements. I really do not know where to start. Can someone give me hint how to prove this?

Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.

I’ve figured out that if I know $G$ is not cyclic, then it for any $a \in G, o(a) \neq 4$ (or the order of any element in group $G$ is not 4). I know ahead of time that the elements in the group ($\forall x \in G$) must have order $o(x)=k$ where $0 < […]

$p(x)$ irreducible polynomial $\iff J=\langle p(x)\rangle$ is a maximal ideal in $K$ $\iff K/J$ is a field

Given a field $K$ and $p(x)\in K[x]$. Then the following conditions are equivalent: a) $p(x)$ is irreducible over $K$. b) $J = \langle p(x)\rangle$ is a maximal ideal in $K[x]$. c) $K[x]/J$ is a field, where $J=\langle p(x)\rangle$. My book proves $a\implies b$ as follows: Since the degree of $p(x)$ is greater than or equal […]

Is there a commutative operation for which the inverse of the operation is also commutative?

For instance addition is commutative, but the inverse, subtraction, is not. $$ 5+2 = 2+5\\ 5-2 \neq 2-5 $$ Same for multiplication/division: $$ 5\times4 = 4\times5\\ 5/4 \neq 4/5 $$ So is there a group operation $\circ$ with the inverse $\circ^{-1}$ such that $$ a\circ b = b\circ a\\ a\circ^{-1}b = b\circ^{-1}a $$

Generalization of index 2 subgroups are normal

Let $G$ be a finite group and $H$ a subgroup of index $p$, where $p$ is a prime. If $\operatorname{gcd}(|H|, p-1)=1$, then $H$ must be normal. Does somebody have a quick proof of this?

Quotient of a Clifford algebra by its radical is a Clifford algebra?

I’m fumbling a bit in my reading on Clifford algebras. I’m hoping someone can shed some light on the following isomorphism. Suppose you have a symmetric bilinear form $G$ over a vector space $V$, and let $\mathrm{Cl}_G(V)$ be the corresponding Clifford algebra. I’ll denote it by $C_G$ for short when the vector space is clear. […]

Zero divisor in $R$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there’s an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=ba_n=0$?

Order of a product of subgroups

Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$. I need this theorem to prove something.

Proof of an alternative form of Fermat-Euler's theorem.

I want to know a proof of an alternative form of Fermat-Euler’s theorem $$a^{\phi (n) +1} \equiv a \pmod n$$ when $a$ and $n$ are not relatively prime. I searched some number theory books and a cryptography book and internet, but there were only proofs of the original theorem $a^{\phi (n)} \equiv 1 \pmod n$ […]

The ring $ℤ/nℤ$ is a field if and only if $n$ is prime

Let $n \in ℕ$. Show that the ring $ℤ/nℤ$ is a field if and only if $n$ is prime. Let $n$ prime. I need to show that if $\bar{a} \neq 0$ then $∃\bar b: \bar{a} \cdot \bar{b} = \bar{1}$. Any hints for this ? Suppose $ℤ/nℤ$ is a field. Therefore: for every $\bar{a} \neq 0$ […]