A ring extension S of a ring
R is right intrinsic over R, in the sense of Faith and Utumi, if A ∩ R≠0 for each
nonzero right ideal A of S. S is a right quotient ring of R, in the sense of R. E.
Johnson, if SR is an essential extension of RR. Let 𝒞 be the class of prime rings
which have zero right singular ideal and contain uniform right ideals. This paper
deals with two questions:
If S ∈𝒞 and S is right intrinsic over a prime ring R, is S a right quotient
ring of R?
If R ∈𝒞 and S is right intrinsic over R, is S a right quotient ring of R?
The main result is that the answer to (1) is “yes” provided S is not an integral domain.
As a consequence of this, a partial answer to (2) is “yes” provided R is not an
integral domain and R contains a nonzero finite dimensional right annihilator
ideal.
