Abstract
Let N be a completely prime ideal of a right distributive ring A, and let \( N \subseteq J(A). \) Then the following assertions hold.
-
(1)
N = mN ⊂ mA for all m ∈ A\N, and A has no nontrivial idempotents.
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(2)
N is comparable to any right ideal of A.
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(3)
M N = N for any right ideal M of A which is not contained in N.
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(4)
Either N = 0 and A is a right uniform domain, or N is a nonzero essential right ideal of A.
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(5)
Either the module NA IS not uniform, or A is right uniform.
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(6)
If N is a finitely generated left ideal, then either N = J(A), or N = 0 and A is a right uniform domain.
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© 1998 Springer Science+Business Media Dordrecht
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Tuganbaev, A.A. (1998). Distributive rings with maximum conditions. In: Semidistributive Modules and Rings. Mathematics and Its Applications, vol 449. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5086-6_9
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DOI: https://doi.org/10.1007/978-94-011-5086-6_9
Publisher Name: Springer, Dordrecht
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