Ideal (ring theory): Difference between revisions

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In [[ring theory]], a branch of [[abstract algebra]], an '''ideal''' of a [[ring (algebra)|ring]] <i>R</i> is a [[subset]] <i>I</i> of <i>R</i> which is closed under <i>R</i>-[[linear combination]]s,. inThe aterm sense''ideal'' madecomes precisefrom below.the The conceptnotion of an'''ideal [[ordernumber''' ideal]]and thatgeneralizes isthe knownconcept inof a [[order theorynumber]]. is derived from this notion is discussed in its dedicated article.

The term "ideal" comes from the notion of '''ideal number''': ideals were seen as a generalization of the concept of [[number]].

In the ring <b>Z</b> of integers, every ideal can be generated by a single number (so <b>Z</b> is a [[principal ideal ___domain]]), and the ideal determines the number up to its sign.

The concepts of "ideal" and "number" are therefore almost identical in <b>Z</b> (and in any principal ideal ___domain).

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In a certain class of rings important in [[number theory]], the [[Dedekind ___domain]]s, one can even recover a version of the [[fundamental theorem of arithmetic]]: in these rings, every nonzero ideal can be uniquely written as a product of prime ideals.

An ideal can be used to construct a factor ring in a similar way as a [[normal subgroup]] in [[group theory]] can be used to construct a [[factor group]]. The concept of an [[order ideal]] in [[order theory]] is derived from this notion.

== Definitions ==