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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

Kevin Arana
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  • What have you tried so far? – ckefa Jan 23 '22 at 22:58
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    Are you sure the first bullet isn't to show $A$ is anticommutative? – Eric Towers Jan 23 '22 at 23:06
  • @EricTowers You actually get commutativity and anti-commutativity. – Thomas Andrews Jan 23 '22 at 23:11
  • Each and every question in this list is answered on the site already, and not hard to find. Even your entire list itself is a duplicate. In general, please do not ask lists of questions, unless they are highly interrelated so that asking them separately doesn't work well. – rschwieb Jan 24 '22 at 16:41
  • Oh, somehow I didn't explicitly write out the advice to please search first. When i did the search, I just did "boolean domain maximal generated" and got the duplicates I linked, and many things linked to them. – rschwieb Jan 24 '22 at 16:48

1 Answers1

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  • 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 $\cap$.
Alessandro
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