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The $k$-Closure of Monic and Monic Free Ideals in a Polynomial Semiring

Louis Dale
Proceedings of the American Mathematical Society
Vol. 64, No. 2 (Jun., 1977), pp. 219-226
DOI: 10.2307/2041431
Stable URL: http://www.jstor.org/stable/2041431
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The $k$-Closure of Monic and Monic Free Ideals in a Polynomial Semiring
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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 $S\lbrack x \rbrack$ satisfies the ascending chain condition for monic ideals.

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