Distributive rings with maximum conditions
N = mN ⊂ mA for all m ∈ A\N, and A has no nontrivial idempotents.
N is comparable to any right ideal of A.
M N = N for any right ideal M of A which is not contained in N.
Either N = 0 and A is a right uniform domain, or N is a nonzero essential right ideal of A.
Either the module NA IS not uniform, or A is right uniform.
If N is a finitely generated left ideal, then either N = J(A), or N = 0 and A is a right uniform domain.
KeywordsPrime Ideal Left Ideal Division Ring Noetherian Ring Semiprime Ring
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