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Let R be a ring and S a subring
of R. Let ω be a ring homomorphism mapping S onto a division ring Γ. Choose an
ideal P ⊆ R maximal with respect to the property (P ∩S)ϕ = (0). P is a prime ideal
of R. If P is any prime ideal of R which can be obtained in the above manner write
P = P(Γ,S,φ).
It is shown that a# l primitive ideals are of the form P = P(Γ,S,φ) and that a
ring R is nil if and only if it has no prime ideals of the form P = P(Γ,S,φ). It is
shown that the nil radica# of any ring is the intersection of all prime ideals
P = P(Γ,S,φ).
It is shown that if P = P(Γ,,S,φ) for all prime ideals P ⊆ R then the nil and
Baer radicals coincide for all homomorphic images of R. If the nil and Baer radicals
coincide for all homomorphic images of R, it is shown that any prime ideal P of R is
contained in a prime ideal P′ = Pγ(Γ,S,φ).
Finally, by consideration of prime ideals P = P(Γ,S,φ), two theorems are proved giving
information about rings satisfying very special conditions.
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