Krull dimension in power series rings

Author:
Jimmy T. Arnold

Journal:
Trans. Amer. Math. Soc. **177** (1973), 299-304

MSC:
Primary 13J05

MathSciNet review:
0316451

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *R* denote a commutative ring with identity. If there exists a chain of prime ideals of *R*, where , but no such chain of prime ideals, then we say that *R* has dimension *n*. The power series ring may have infinite dimension even though *R* has finite dimension.

**[1]**J. T. Arnold and J. W. Brewer,*When (𝐷[[𝑋]])_{𝑃[[𝑋]]} is a valuation ring*, Proc. Amer. Math. Soc.**37**(1973), 326–332. MR**0311656**, 10.1090/S0002-9939-1973-0311656-X**[2]**David E. Fields,*Zero divisors and nilpotent elements in power series rings*, Proc. Amer. Math. Soc.**27**(1971), 427–433. MR**0271100**, 10.1090/S0002-9939-1971-0271100-6**[3]**David E. Fields,*Dimension theory in power series rings*, Pacific J. Math.**35**(1970), 601–611. MR**0277518****[4]**Robert W. Gilmer,*Multiplicative ideal theory*, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR**0229624****[5]**Jack Ohm and R. L. Pendleton,*Rings with noetherian spectrum*, Duke Math. J.**35**(1968), 631–639. MR**0229627****[6]**A. Seidenberg,*A note on the dimension theory of rings*, Pacific J. Math.**3**(1953), 505–512. MR**0054571****[7]**A. Seidenberg,*On the dimension theory of rings. II*, Pacific J. Math.**4**(1954), 603–614. MR**0065540**

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0316451-8

Keywords:
Dimension,
power series ring

Article copyright:
© Copyright 1973
American Mathematical Society