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meet-irreducible, but rather satisfy the following condition: R is not zero, and if the intersection of two ideals of R is nil, then one of the ideals is nil. (An ideal is called nil if every element is nilpotent.) Also, it is a bit confusing that this condition is mentioned in the section with the definitions, but not in the introduction, and that it has no name. What about "spectrally irreducible" and/or also including the description given above? Of course, in algebraic geometry, one just says "irreducible". --
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The sentence "Commutative meet-irreducible rings play an elementary role in algebraic geometry, where this concept is generalized to the concept of an irreducible scheme." in the last section is not really precise, since irreducible affine schemes do not correspond to commutative rings R which are
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The four conditions for irreducible commutative rings are not equivalent. For example, the quotient ring
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is a field, has a prime (in fact, maximal) nilradical but the intersection of (
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Noetherian ring that is not irreducible. (See
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