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<div>In [[mathematics]], a '''tertiary ideal''' is an (two-sided) [[Ideal (ring theory)|ideal]] in a (perhaps noncommutative) [[Ring (mathematics)|ring]] that cannot be expressed as a nontrivial intersection of a right [[fractional ideal]] with another ideal. Tertiary ideals generalize [[primary ideal]]s to the case of [[noncommutative ring]]s. Although [[primary decomposition]]s do not exist in general for ideals in noncommutative rings, tertiary decompositions do, at least if the ring is [[Noetherian ring|Noetherian]].<br />
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Every primary ideal is tertiary. Tertiary ideals and primary ideals coincide for commutative rings. To any (two-sided) ideal, a tertiary ideal can be associated called the tertiary radical, defined as<br />
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:<math>t(I) = \{r \in R \mbox{ }|\mbox{ } \forall s \notin I, \mbox{ }\exists x \in (s)\mbox{ } x \notin I \text{ and } (x)(r) \subset I \}. \, </math><br />
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Then ''t''(''I'') always contains ''I''.<br />
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If ''R'' is a (not necessarily commutative) Noetherian ring and ''I'' a right ideal in ''R'', then ''I'' has a unique irredundant decomposition into tertiary ideals<br />
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:<math>I = T_1 \cap \dots \cap T_n</math>.<br />
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== See also ==<br />
* [[Primary ideal]]<br />
* [[Lasker–Noether theorem]]<br />
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== References ==<br />
* {{Citation | last=Riley | first=J.A. | title=Axiomatic primary and tertiary decomposition theory | journal=Trans. Amer. Math. Soc. | year=1962 | volume=105 | pages=117–201}}<br />
* [http://www.encyclopediaofmath.org/index.php/Tertiary_ideal Tertiary ideal], Encyclopedia of Mathematics, Springer Online Reference Works.<br />
* {{Citation | last=Behrens | first=Ernst-August | title=Ring Theory | publisher=Verlag Academic Press | year=1972 | url=http://books.google.ch/books?id=ZKGq4IQHhHUC&lpg=PP1&pg=PP1#v=onepage&q=&f=false}}<br />
* {{Citation | last=Kurata | first=Yoshiki | title=On an additive ideal theory in a non-associative ring | journal=Mathematische Zeitschrift | volume=88 | issue=2 | year=1965 | doi=10.1007/BF01112095 | pages=129–135 | url=http://www.springerlink.com/content/h772w68514700345/}}<br />
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{{algebra-stub}}<br />
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{{DEFAULTSORT:Tertiary ideals}}<br />
[[Category:Algebra]]</div>216.65.182.66