The Journal of the Indian Mathematical Society

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Local Nullstellensatz over Commutative Ground Rings


Affiliations
1 Department of Mathematics, Missouri State University, Springfield, Missouri 65897, United States
2 Department of Mathematics, University of California, Riverside, California 92521-0135, United States
     

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It is shown that a local Nullstellensatz holds over an arbitrary commutative ring A (with identity 1 ≠ 0); specifically, if B = A[x1, . . . , xn] is a finitely generated extension ring of A and N is a maximal ideal in B, then NBN = (N ∩ A, x1 − c1, . . . , xn − cn)BN for some c1, . . . , cn ∈ BN .


Keywords

G-Ideal, Nullstellensatz, Maximal Ideal, Polynomial Ring.
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  • D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995.

  • I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.

  • P. Kemp, L. J. Ratliff, Jr., and K. Shah, Depth one homogeneous prime ideals in polynomial rings over a field, Journal of Indian Math. Soc. (accepted).

  • M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.

  • O. Zariski and P. Samuel, Commutative Algebra, Vol. 1, D. Van Nostrand, New York, 1958.

  • O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, D. Van Nostrand, New York, 1960.


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  • Local Nullstellensatz over Commutative Ground Rings

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Authors

Paula Kemp
Department of Mathematics, Missouri State University, Springfield, Missouri 65897, United States
Louis J. Ratliff
Department of Mathematics, University of California, Riverside, California 92521-0135, United States
Kishor Shah
Department of Mathematics, Missouri State University, Springfield, Missouri 65897, United States

Abstract


It is shown that a local Nullstellensatz holds over an arbitrary commutative ring A (with identity 1 ≠ 0); specifically, if B = A[x1, . . . , xn] is a finitely generated extension ring of A and N is a maximal ideal in B, then NBN = (N ∩ A, x1 − c1, . . . , xn − cn)BN for some c1, . . . , cn ∈ BN .


Keywords


G-Ideal, Nullstellensatz, Maximal Ideal, Polynomial Ring.

References