Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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THE S-FINITENESS ON QUOTIENT RINGS OF A POLYNOMIAL RING

  • LIM, JUNG WOOK (Department of Mathematics, College of Natural Sciences, Kyungpook National University) ;
  • KANG, JUNG YOOG (Department of Mathematics Education, Silla University)
  • Received : 2021.03.01
  • Accepted : 2021.03.03
  • Published : 2021.09.30
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Abstract

Let R be a commutative ring with identity, R[X] the polynomial ring over R and S a multiplicative subset of R. Let U = {f ∈ R[X] | f is monic} and let N = {f ∈ R[X] | c(f) = R}. In this paper, we show that if S is an anti-Archimedean subset of R, then R is an S-Noetherian ring if and only if R[X]U is an S-Noetherian ring, if and only if R[X]N is an S-Noetherian ring. We also prove that if R is an integral domain and R[X]U is an S-principal ideal domain, then R is an S-principal ideal domain.

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Acknowledgement

We would like to thank the referee for several valuable suggestions.

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