# Sum of two prime ideals is prime ?

## Main Question or Discussion Point

i was looking for a counter example.

and, i've not been able to think of any.

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look at Z

mathwonk
Homework Helper
the sum of two ideals is the ideal generated by their union right?

thus geometrically it is the ideal of the intersection of the two zero loci.

so look for a pair of irreducible algebraic sets whose intersection is reducible,

(like a quadric surface and a tangent plane.)

i.e. a prime ideal is one that has a (reduced and) irreducible zero set.

i did not understand the geometric bit.
well i considered the principal ideals <2> and <3>
their union includes 1 which is a unit in Z, so the ideal of the sum is nothing but Z itself right ?
and that can't be prime by definition ? (since an ideal P is prime => P /= R (the ring in consideration))

i did not understand the geometric bit.
well i considered the principal ideals <2> and <3>
their union includes 1 which is a unit in Z, so the ideal of the sum is nothing but Z itself right ?
and that can't be prime by definition ? (since an ideal P is prime => P /= R (the ring in consideration))
yea that's fine, just notice 1 = -1*2 + 1*3

another way to think about it is in terms of existence of gcd's. gcd(2, 3) = 1, so there are x, y in Z such that 2x + 3y = 1 and this is in <2> + <3>, but this is an ideal, so x = x*1 is in <2> + <3> for all x in Z, so yea Z = <2> + <3>, probably overkill but a useful observation

this can be generalized and is really useful, if R is a pid, then Ra + Rb = Rd where d = gcd(a, b)

when looking for counterexamples always think simple(doesn't always work but sometimes it does), like for example 2Z and 3Z are prime but 2Z n 3Z = 6Z is not, so the intersection of prime ideals isn't necessarily prime

Last edited:
Hurkyl
Staff Emeritus
Gold Member
i did not understand the geometric bit.
Think of rings like R[x, y]. Algebraic curves (like the parabola y - x^2 = 0) correspond to ideals (like the ideal <y - x^2>). Sums of ideals relate to intersections of curves. Can you work out why? Do you see how a non-prime ideal corresponds, in some sense, into a curve that is the union of two or more other curves?

mathwonk