Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set: P(A)={x|x sube A} Show that symmetric deference operation on

Let $\times$ be a binary operation on set of rational number ${\displaystyle \mathbb{Q}}$ defined as follows: $a\cdot b=a+b+2ab$, where ${\displaystyle a,b\in \mathbb{Q}}$
a) Prove that $\times$ is commutative, associate algebraic operation on ${\displaystyle \mathbb{Q}}$

Suppose that R is a commutative ring and $|R|=30$. If I is an ideal of R and $|I|=10$, prove that I is a maximal ideal.

If A and B are ideals of a commutative ring R with unity and A+B=R show that ${\displaystyle A\cap B=AB}$

Let $A$ be a domain. Assume that for any non-trivial finitely generated $A$-module $M$ we have ${\mathrm{Hom}}_A(M,A)\ne \{0\}$. Prove that $A$ is a field.

suppose $A$ is a $C^{\ast }$-algebra and $x$ is a normal element in $A$. $C^{\ast }$-algebra generated by $x$ denote by $A[x]$. then
1) $A[x]$ is commutative.
2) $A[x]$ is the clouser of polynomials of two variable $x$ and $x^{\ast }$.

Let's fix the notation, $V=\underset{i\ge 0}{⨁}{V}^i$ is a graded vector space and $\mathrm{\Lambda }V$ is the free commutative graded algebra on $V$. I have been struggling to understand this example:
Consider a graded vector space $V$ with basis $\{a,b\}$ such that $a\in V^2$ and $b\in V^5$. Now define a linear map $d$ (of degree 1) by $da=0$ and $db=a^3$. It follows that d extends uniquely to a derivation .
The point of the example is to show that the derivation on $\mathrm{\Lambda }V$ is completely determined by its values on $V$. So if i understand well, he considers a linear map $d:V⟶\mathrm{\Lambda }V$ of degree one defined by
(here ${\mathrm{\Lambda }}^{k}V$ is the set of elements of word length $k$) and
$d_5:V^5⟶{\mathrm{\Lambda }}^{6}V;b\mapsto a^3$
The first question that i'm stuck on is for $d_2(b)=a^3$, i mean $a^3$ is of length $3$, how it can be in ${\mathrm{\Lambda }}^{6}V$.

Let $R$ be a commutative ring. The first Weyl algebra over $R$ is the associative $R$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$.
or which rings $R$, the first Weyl algebra $A_1(R)$ is simple?