Unit ring non-trivial where every element is idempotent

Question

Let $A$ be a nontrivial unit ring such that $x^2 = x$ for all $x \in A$.

• Calculate $(x+y)^2$ and deduce that A is commutative.
• Prove that if $A$ is domain, then $A \cong \mathbb{Z}_2$.
• Prove that every prime ideal of $A$ is maximal.
• Prove that every ideal $I = (x,y)$ generated by 2 elements is principal. Deduce that every ideal generated by a finite number of elements is principal.[hint: take a cue from this ring $(\mathcal{P}(S),\Delta,\cap)$ calculating the generator of an ideal $(x,y)\subseteq \mathcal{P}(S)$.

Could you give me some suggestions for each of these questions? The question that instead concerns the first ideal .. I have no idea how I can do to demonstrate that the ideal is also maximal

Answer 1

• The question itself is an hint for the solution: Start by calculating $(x+y)^2$ and deduce $xy=-yx$. Then prove $-1=1$ in $A$ and you get commutativity
• If you rewrite $x^2=x$ as $x(1-x)=0$ and $A$ is a domain, what can you deduce about any $x\in A$?
• If $\mathfrak p$ is a prime ideal, or any ideal really, then $A/\mathfrak p$ will also have the property that every element is idempotent. What can you say about $A/\mathfrak p$? The key for the solution is in the previous point
• Suggestion: Work out the example. You'll find out the ideal $(X,Y)$ in $\mathcal P(S)$ is generated by a certain set constructed from $X,Y$, can you describe how this set is constructed using $\Delta$ and $\cap$ on $X,Y$? If yes, the general case has $+$ instead of $\Delta$ and $\cdot$ instead of $\cap$.