ideals contained in a union of ideals

Assume that R is a commutative ring.

Lemma. Let A, B, C be ideals in R such that ABC. Then AB or AC.

Proof. Assume that this is not true. Then there are x,yA such that xB, yC and xC, yB. Obviously x+yABC and without loss of generality we may assume that x+yB. Then y=(x+y)-xB. ContradictionMathworldPlanetmathPlanetmath.

Remark. This lemma is also true if we exchange ring with a group and ideals with subgroupsMathworldPlanetmathPlanetmath (because we didn’t use multiplicationPlanetmathPlanetmath and commutativity of addition in proof).

PropositionPlanetmathPlanetmathPlanetmath. Let I, P1,,Pn be ideals in R such that each Pi is prime. If IP1Pn, then there exists i{1,,n} such that IPi.

Proof. We will use the inductionMathworldPlanetmath on n. For n=2 our lemma applies. Let n>2. Assume that IP1Pn. For i{1,,n} define


By our assumptionPlanetmathPlanetmath (and induction hypothesis) IPi¯ for any i{1,,n}. Thus for any i there is xiI such that xiPi¯.

Now for any i{1,,n} define xi¯=x1xi-1xi+1xnI. Then we have


and thus there is j{1,,n} such that x1¯++xn¯Pj. Since xi¯Pj for any ij, then we have that


But Pj is prime, so there is kj such that xkPjPk¯. Contradiction.

Counterexample. We will show, that if Pi’s are not prime, then the thesis no longer hold, even when n=3. Consider the ring of polynomials in two variables over a simple field of order 2, i.e. 2[X,Y]. Let R=2[X,Y]/(X2,XY,Y2). For W(X,Y)2[X,Y] we shall write W(X,Y)¯=W(X,Y)+(X2,XY,Y2)R. Then it is easy to see, that




It can be easily checked, that I,A1,A2,A3 are all ideals and IA1A2A3 but obviously IAi for any i=1,2,3.

Title ideals contained in a union of ideals
Canonical name IdealsContainedInAUnionOfIdeals
Date of creation 2013-03-22 19:03:55
Last modified on 2013-03-22 19:03:55
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 13A15
Related topic IdealIncludedInUnionOfPrimeIdeals