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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Monic and monic free ideals in a polynomial semiring in several variables


Author: Louis Dale
Journal: Proc. Amer. Math. Soc. 61 (1976), 209-216
MSC: Primary 16A66
DOI: https://doi.org/10.1090/S0002-9939-1976-0427383-7
MathSciNet review: 0427383
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Abstract: The study of monic and monic free ideals in a polynomial semiring $ S[x]$, where $ S$ is a commutative semiring with an identity, is extended to $ S[{x_1},{x_2}, \ldots ,{x_n}]$. Structure theorems are given for monic, monic free, and monic free $ k$-ideals in $ S[{x_1},{x_2}, \ldots ,{x_n}]$. It is shown that each monic free $ k$-ideal in $ S[{x_1}, \ldots ,{x_n}],\;S$ a strict semiring, is the sum of a finite number of ideals $ {B_i}$ such that each $ {B_i}$ is the union of a proper infinite ascending chain of ideals.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0427383-7
Keywords: Strict semiring, monic ideals, monic free ideals, $ k$-ideals
Article copyright: © Copyright 1976 American Mathematical Society