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The $ k$-closure of monic and monic free ideals in a polynomial semiring


Author: Louis Dale
Journal: Proc. Amer. Math. Soc. 64 (1977), 219-226
MSC: Primary 16A78
DOI: https://doi.org/10.1090/S0002-9939-1977-0444717-9
MathSciNet review: 0444717
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Abstract: The concepts of k-closure, k-boundary and weak k-ideals are introduced and necessary and sufficient conditions that an ideal be a k-ideal are given. These conditions are applied to monic and monic free k-ideals. Also, it is shown that the ascending chain condition holds for monic ideals, but not for monic free ideals, and that a semiring S is Noetherian if and only if $ S[x]$ satisfies the ascending chain condition for monic ideals.


References [Enhancements On Off] (What's this?)

  • [1] L. Dale, Monic and monic free ideals in a polynomial semiring, Proc. Amer. Math. Soc. 56 (1976), 45-50. MR 0404354 (53:8156)
  • [2] L. Dale and P. J. Allen, Ideal theory in the semiring $ {Z^ + }$, Publ. Math. Debrecen 22 (1975), 219-224. MR 0404352 (53:8154)
  • [3] P. J. Allen, A fundamental theorem of homomorphisms for semirings, Proc. Amer. Math. Soc. 21 (1969), 412-416. MR 0237575 (38:5856)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0444717-9
Keywords: Strict semiring, monic ideals, monic free ideals, k-ideals, weak k-ideals, k-closure, k-boundary
Article copyright: © Copyright 1977 American Mathematical Society

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