A homogeneous ideal that is homogeneously prime but not prime #

In Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem, we assumed that the underlying grading is indexed by a LinearOrderedCancelAddCommMonoid to prove that a homogeneous ideal is prime if and only if it is homogeneously prime. This file shows that even if this assumption isn't strictly necessary, the assumption of "being cancellative" is. We construct a counterexample where the underlying indexing set is a LinearOrderedAddCommMonoid but is not cancellative and the statement is false.

We achieve this by considering the ring R=ℤ/4ℤ. We first give the two element set ι = {0, 1} a structure of linear ordered additive commutative monoid by setting 0 + 0 = 0 and _ + _ = 1 and 0 < 1. Then we use ι to grade by setting {(a, a) | a ∈ R} to have grade 0; and {(0, b) | b ∈ R} to have grade 1. Then the ideal I = span {(2, 2)} ⊆ ℤ/4ℤ × ℤ/4ℤ is homogeneous and not prime. But it is homogeneously prime, i.e. if (a, b), (c, d) are two homogeneous elements then (a, b) * (c, d) ∈ I implies either (a, b) ∈ I or (c, d) ∈ I.

Tags #

homogeneous, prime

The grade 0 part of is {(a, a) | a ∈ R}.

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    The grade 1 part of is {(0, b) | b ∈ R}.

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      Given the above grading (see submoduleZ and submoduleO), we turn into a graded ring.

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        The counterexample is the ideal I = span {(2, 2)}.

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