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Let $R$ be a strongly $2$-primal ring and $I$ a proper ideal of $R$. Then there are only finitely many prime ideals minimal over $I$ if and only if for every prime ideal $P$ minimal over $I$, the ideal $P/\sqrt{I}$ of $R/\sqrt{I}$ is finitely generated if and only if the ring $R/\sqrt{I}$ satisfies the \emph{ACC} on right annihilators. This result extends ``D. D. Anderson, A note on minimal prime ideals, \emph{Proc. Amer. Math. Soc.} 122 (1994), no. 1, 13--14." to large classes of noncommutative rings. It is also shown that, a $2$-primal ring $R$ only has finitely many minimal prime ideals if each minimal prime ideal of $R$ is finitely generated. Examples are provided to illustrate our results.
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참고문헌 (18)
참고문헌 신청
A. H. Al-Huzali / 1992 / Rings whose cyclics have finite Goldie dimension / J. Algebra 153(1):37~40
D. D. Anderson / 1994 / A note on minimal prime ideals / Proc. Amer. Math. Soc 122(1):13~14
G. F. Birkenmeier / 1993 / Ring Theory / World Scientific :102~129
V. P. Camillo / 1977 / Modules whose quotients have finite Goldie dimension / Pacific J. Math 69(2):337~338
A. W. Chatters / 1980 / Rings with Chain Conditions / Pitman Advanced Publishing Program
D. D. Anderson / 1994 / A note on minimal prime ideals / Proc. Amer. Math. Soc 122(1):13~14
G. F. Birkenmeier / 1993 / Ring Theory / World Scientific :102~129
V. P. Camillo / 1977 / Modules whose quotients have finite Goldie dimension / Pacific J. Math 69(2):337~338
A. W. Chatters / 1980 / Rings with Chain Conditions / Pitman Advanced Publishing Program
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