A Sufficient Condition for a Minimal Ring Extension to Be an Overring

REFERENCES

• Anderson , D. F. , Badawi , A. , Dobbs , D. E. ( 1997 ). Pseudo-Valuation Rings . Lecture Notes Pure Appl. Math. 185. New York/Basel : Dekker , pp. 57 – 67 .
   Google Scholar
• Anderson , D. D. , Dobbs , D. E. , Mullins , B. ( 1999 ). The primitive element theorem for commutative algebras . Houston J. Math. 25 : 603 – 623 .
   Web of Science ®Google Scholar
• Anderson , D. F. , Badawi , A. , Dobbs , D. E. ( 2000 ). Pseudo-valuation rings, II . Boll. U. M. I. 8 ( 3-B ): 535 – 545 .
   Google Scholar
• Atiyah , M. , Macdonald , I. G. ( 1969 ). Introduction to Commutative Algebra . Reading : Addison-Wesley .
   Google Scholar
• Dechéne , L. I. ( 1978 ). Adjacent Extensions of Rings . Ph. D. dissertation, University of California, California, USA, Riverside .
   Google Scholar
• Dobbs , D. E. ( 1997 ). Going-down rings with zero-divisors . Houston J. Math. 23 : 1 – 12 .
   Web of Science ®Google Scholar
• Dobbs , D. E. ( 2006 ). Every commutative ring has a minimal ring extension . Comm. Algebra 34 ( 10 ): 3875 – 3881 .
   Web of Science ®Google Scholar
• Dobbs , D. E. , Shapiro , J. ( 2006 ). A classification of the minimal ring extensions of an integral domain . J. Algebra 305 : 185 – 193 .
   Web of Science ®Google Scholar
• Ferrand , D. , Olivier , J.-P. ( 1970 ). Homomorphismes minimaux d'anneaux . J. Algebra 16 : 461 – 471 .
   Web of Science ®Google Scholar
• Fontana , M. ( 1980 ). Topologically defined classes of commutative rings . Ann. Mat. Pura Appl. 123 : 331 – 355 .
   Google Scholar
• Gilbert , M. S. ( 1996 ). Extensions of Commutative Rings with Linearly Ordered Intermediate Rings . Ph. D. dissertation, University of Tennessee, Knoxville .
   Google Scholar
• Gilmer , R. ( 1972 ). Multiplicative Ideal Theory . New York : Dekker .
   Google Scholar
• Hedstrom , J. R. , Houston , E. G. (1978). Pseudo-valuation domains. Pac. J. Math. 75:137–147.
   Web of Science ®Google Scholar
• Kaplansky , I. ( 1974 ). Commutative Rings. , Rev. Ed. Chicago : University of Chicago Press .
   Google Scholar
• Matsuda , R. ( 1992 ). Remarks on Sato-Sugatani-Yochida's results concerning minimal overrings . Math. J. Toyama Univ. 15 : 39 – 42 .
   Google Scholar
• Nagata , M. ( 1962 ). Local Rings . New York : Wiley-Interscience .
   Google Scholar
• Picavet , G. , Picavet-L'Hermitte , M. About minimal morphisms . Multiplicative Ideal Theory: A Tribute to the Work of Robert Gilmer . Springer-Verlag , to appear .
   Google Scholar
• Sato , J. , Sugatani , T. , Yochida , K. I. ( 1992 ). On minimal overrings of a Noetherian domain . Comm. Algebra 20 : 1735 – 1746 .
   Web of Science ®Google Scholar
• Zariski , O. , Samuel , P. ( 1958 ). Commutative Algebra . Vol. I. Princeton : Van Nostrand .
   Google Scholar
• Communicated by J. Kuzmanovich.
   Google Scholar