Geodesic
On the Prime Ideals in a Commutative Ring
Canadian mathematical bulletin, Tome 43 (2000) no. 3, pp. 312-319

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If $n$ and $m$ are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring $R$ with exactly $n$ elements and exactly $m$ prime ideals. Next, assuming the Axiom of Choice, it is proved that if $R$ is a commutative ring and $T$ is a commutative $R$ -algebra which is generated by a set $I$ , then each chain of prime ideals of $T$ lying over the same prime ideal of $R$ has at most ${{2}^{\left| I \right|}}$ elements. A polynomial ring example shows that the preceding result is best-possible.
DOI : 10.4153/CMB-2000-038-7
Mots-clés : 13C15, 13B25, 04A10, 14A05, 13M05 
Dobbs, David E. On the Prime Ideals in a Commutative Ring. Canadian mathematical bulletin, Tome 43 (2000) no. 3, pp. 312-319. doi: 10.4153/CMB-2000-038-7
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     title = {On the {Prime} {Ideals} in a {Commutative} {Ring}},
     journal = {Canadian mathematical bulletin},
     pages = {312--319},
     year = {2000},
     volume = {43},
     number = {3},
     doi = {10.4153/CMB-2000-038-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2000-038-7/}
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