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An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924401.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924402.png" /> which cannot be expressed as the intersection of a right [[Fractional ideal|fractional ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924403.png" /> and an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924404.png" />, each strictly larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924405.png" />. All irreducible ideals are tertiary. In Noetherian rings, tertiary ideals are the same as primary ideals (cf. [[Additive theory of ideals|Additive theory of ideals]]; [[Primary ideal|Primary ideal]]; [[Primary decomposition|Primary decomposition]]).
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An ideal $I$ of a ring $R$ which cannot be expressed as the intersection of a right [[fractional ideal]] $r(I,A)$ and an ideal $B$, each strictly larger than $I$. All [[irreducible ideal]]s are tertiary. In [[Noetherian ring]]s, tertiary ideals are the same as [[primary ideal]]s (cf. [[Additive theory of ideals]]; [[Primary decomposition]]).
  
Suppose that the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924406.png" /> satisfies the maximum condition for left and right fractional ideals, and that every ideal decomposes as an intersection of finitely many indecomposable ideals. Then for every ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924407.png" /> there exists a tertiary radical, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924408.png" />, the largest ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t0924409.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t09244010.png" /> such that, for any ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t09244011.png" />,
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Suppose that the ring $R$ satisfies the maximum condition for left and right fractional ideals, and that every ideal decomposes as an intersection of finitely many indecomposable ideals. Then for every ideal $Q$ there exists a tertiary radical, $\mathrm{ter}(Q)$, the largest ideal $T$ of $R$ such that, for any ideal $B$,
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t09244012.png" /></td> </tr></table>
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r(Q,T) \cap B = Q \ \Rightarrow\  B=Q \ .
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$$
  
 
As for primary ideals, the intersection theorem, the existence theorem and the uniqueness theorem are true for tertiary ideals.
 
As for primary ideals, the intersection theorem, the existence theorem and the uniqueness theorem are true for tertiary ideals.
  
An analysis of the properties of left and right fractions (of ideals of a ring, of submodules of a module, and others) leads to systems with fractions in which the general notions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t09244013.png" />-primarity and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092440/t09244014.png" />-primary radicals occur naturally. This allows one to formulate the  "intersection" ,  "existence"  and  "uniqueness theorems"  as axioms. In this approach, tertiarity is the unique notion of primarity for which all these three theorems hold, i.e. it is the unique  "good"  generalization of classical primarity (cf. [[#References|[1]]], [[#References|[2]]]).
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An analysis of the properties of left and right fractions (of ideals of a ring, of submodules of a module, and others) leads to systems with fractions in which the general notions of $S$-primarity and $S$-primary radicals occur naturally. This allows one to formulate the  "intersection" ,  "existence"  and  "uniqueness theorems"  as axioms. In this approach, tertiarity is the unique notion of primarity for which all these three theorems hold, i.e. it is the unique  "good"  generalization of classical primarity (cf. [[#References|[1]]], [[#References|[2]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Andrunakievich,  Yu.M. Ryabukhin,  "The additive theory of ideals in systems with residuals"  ''Math. USSR Izv.'' , '''1''' :  5  (1967)  pp. 1011–1040  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31''' :  5  (1967)  pp. 1057–1090</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Riley,  "Axiomatic primary and tertiary decomposition theory"  ''Trans. Amer. Math. Soc.'' , '''105'''  (1962)  pp. 117–201</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Andrunakievich,  Yu.M. Ryabukhin,  "The additive theory of ideals in systems with residuals"  ''Math. USSR Izv.'' , '''1''' :  5  (1967)  pp. 1011–1040  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31''' :  5  (1967)  pp. 1057–1090</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Riley,  "Axiomatic primary and tertiary decomposition theory"  ''Trans. Amer. Math. Soc.'' , '''105'''  (1962)  pp. 117–201</TD></TR>
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</table>
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Latest revision as of 19:23, 5 October 2017

An ideal $I$ of a ring $R$ which cannot be expressed as the intersection of a right fractional ideal $r(I,A)$ and an ideal $B$, each strictly larger than $I$. All irreducible ideals are tertiary. In Noetherian rings, tertiary ideals are the same as primary ideals (cf. Additive theory of ideals; Primary decomposition).

Suppose that the ring $R$ satisfies the maximum condition for left and right fractional ideals, and that every ideal decomposes as an intersection of finitely many indecomposable ideals. Then for every ideal $Q$ there exists a tertiary radical, $\mathrm{ter}(Q)$, the largest ideal $T$ of $R$ such that, for any ideal $B$, $$ r(Q,T) \cap B = Q \ \Rightarrow\ B=Q \ . $$

As for primary ideals, the intersection theorem, the existence theorem and the uniqueness theorem are true for tertiary ideals.

An analysis of the properties of left and right fractions (of ideals of a ring, of submodules of a module, and others) leads to systems with fractions in which the general notions of $S$-primarity and $S$-primary radicals occur naturally. This allows one to formulate the "intersection" , "existence" and "uniqueness theorems" as axioms. In this approach, tertiarity is the unique notion of primarity for which all these three theorems hold, i.e. it is the unique "good" generalization of classical primarity (cf. [1], [2]).

References

[1] V.A. Andrunakievich, Yu.M. Ryabukhin, "The additive theory of ideals in systems with residuals" Math. USSR Izv. , 1 : 5 (1967) pp. 1011–1040 Izv. Akad. Nauk SSSR Ser. Mat. , 31 : 5 (1967) pp. 1057–1090
[2] J.A. Riley, "Axiomatic primary and tertiary decomposition theory" Trans. Amer. Math. Soc. , 105 (1962) pp. 117–201
How to Cite This Entry:
Tertiary ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tertiary_ideal&oldid=13632
This article was adapted from an original article by V.A. Andrunakievich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article