Algebra extension: Difference between revisions
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#REDIRECT [[Subring]] |
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In [[mathematics]], more specifically in [[ring theory]], a '''ring extension''' or '''extension ring''' is a [[ring (mathematics)|ring]] ''R'' with a [[subring]] ''S''. We write ''R''/''S'' and say ''R'' is a ''ring extension'' of ''S'' |
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Given an extension ''R''/''S'' of commutative rings and a [[prime ideal]] '''P''' of ''R'', it follows that |
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the intersection, say '''p''', of '''P''' with ''S'' is a prime ideal of ''S''. In this case we say that '''P''' '''lies over''' '''p'''. The situation is more complicated when ''R'' is not commutative. |
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== Examples == |
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* A [[field extension]] is a special case of ring extension. Polynomial ring R[x] is an extension of R. |
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== See also == |
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* [[Integral extension]] |
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* [[Group extension]] |
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* [[Algebraic extension]] |
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* [[Ore extension]] |
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==External links== |
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*{{MathWorld |title=Extension Ring |id=ExtensionRing}} |
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[[Category:Ring theory]] |
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{{Abstract-algebra-stub}} |
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Revision as of 22:09, 4 November 2014
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