Vol. 58, No. 1, 1975

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On one-sided prime ideals

Friedhelm Hansen

Vol. 58 (1975), No. 1, 79–85

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.

Mathematical Subject Classification
Primary: 16A46
Received: 8 April 1974
Published: 1 May 1975
Friedhelm Hansen