Is the sum of two prime ideals prime too?

Question

Let R be a commutative ring with a unity. Suppose that I and J are two prime ideals of this ring, i.e. R/I and R/J are integral domains. Assume that I+J is not the whole ring R. Is it then true that I+J is prime as well? If not, can you find a counterexample?

Answer 1

A good question:
I retract my previous statement - the answer is no. I just consulted with a former professor of mine (Whose name I shall leave nameless but I paraphrase him below) - these counter examples are very tough to generate though -

R = C[x,y]
P=(x^2+y^2-1)
Q=(x)
Both are prime since they are principal ideals generated by irreducible polynomials in a UFD
P+Q = (x^2+y^2-1,x)=(y^2-1,x)
R/(P+Q)=C[x,y]/(y^2-1,x)=C[y]/(y^2-1)
Which clearly has zero divisors y+/-1
So the quotient is not a domain hence the sum is not prime.

Indeed there are some things you need to prove but this is a counter example.

There is also some basic underlying geometry here - one of those ideals above is "the unit circle" the other just the y-axis - These are geometrically irreducible - they do not decompose - however the intersection is two isolated points - which is "geometrically separable". I didn't define much of the above terms however at least you can get a feel for why this works. (Again I credit my professor for his help - most if not all of the ideas here are his and I am paraphrasing his expertise.)

Hope this helps!

So again I credit my professor for giving me the insight here -

Answer 2

I'm not familiar with the term "Prime Ideals" and frankly I can't follow your example but:

Adding any two prime numbers will produce an even number unless one of the primes is "2". By producing an even number it won't be prime. All even numbers are divisible by "2" with a resulting whole number.

That may in no way address your question but it's about the only thing I could offer that might help. Good luck. I hope you get the perfect answer.

Answer 3

Jesus had 12 disciples. Jesus was considered the 13th disciple and was crucified on Friday - ergo: Friday the 13th is bad luck. This also accounts for the number 13 being unlucky. By the way, next Friday happens to be Friday the 13th and it falls in October (Halloween Month). Have a lovely rest of the day.