This paper contains some
results on prime right ideals in weakly regular rings, especially V -rings, and in rings
with restricted minimum condition. Theorem 1 gives information about the structure
of V -rings: A V -ring with maximum condition for annihilating left ideals is a finite
direct sum of simple V -rings. A characterization of rings with restricted minimum
condition is given in Theorem 2: A nonprimitive right Noetherian ring satisfies the
restricted minimum condition iff every critical prime right ideal ≠(0) is maximal. The
proof depends on the simple observation that in a nonprimitive ring with
restricted minimum condition all prime right ideals ≠(0) contain a (two-sided)
prime ideal ≠(0). An example shows that Theorem 2 is not valid for right
Noetherian primitive rings. The same observation on nonprimitive rings leads to a
sufficient condition for rings with restricted minimum condition to be right
Noetherian.
