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I have a simple doubt. Is zero ring a Boolean ring? ( We do have $0 ^2 =0$ ). Or do we assume a Boolean ring to be non-zero?. The doubt came in my mind while showing that in a Boolean ring every prime ideal is maximal.

Mohit Sharma
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There is no requirement that a Boolean ring have more than one element. (Contrast this with fields, where we explicitly require that the additive and multiplicative identities be distinct). So yes, the trivial ring is a Boolean ring.

Re: your edit, note that "prime = maximal" is (appropriately enough) trivially true for the trivial Boolean ring: both prime and maximal ideals are required to be nonempty proper subsets of the ring itself, and the trivial Boolean ring has no nonempty proper subsets at all.

(Actually, it looks like there's a subtlety here: being a proper subset of the ring is not included in the wikipedia definition of prime ideals in the context of noncommutative rings. I suspect that this is an omission on wikipedia's part, but if not then "prime = maximal" using this notion of "prime ideal" does indeed fail for the trivial ring.)

Noah Schweber
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  • Thank you but is not the zero ideal in $\mathbb{Z}$ a prime ideal? – Mohit Sharma Dec 23 '20 at 17:19
  • @MohitSharma What is "$\mathbb{Z}$" here? (Note that that symbol usually denotes the integers, which do not form a Boolean ring.) – Noah Schweber Dec 23 '20 at 17:20
  • If by "$\mathbb{Z}$" you mean the trivial ring, then no, the zero ideal is not prime in the trivial ring since - as I say in my answer - the zero ideal is not a proper subset of the trivial ring. – Noah Schweber Dec 23 '20 at 17:21
  • Oh yes, thank you. I understood. The zero ideal is not a proper subset of the zero ring and the definition of prime ideal requires it to be a proper subset of the ring. – Mohit Sharma Dec 23 '20 at 17:24
  • Sorry i asked about the zero ideal being prime in $\mathbb{Z} $ because in your answer I read non-empty to be non-zero. My mistake!. – Mohit Sharma Dec 23 '20 at 17:34