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S. Carnahan
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The original answer here was incorrect and I removed it. There is a different answer "usingAnother example using idealization construction" submitted with this questionconstruction (i.e. $(a,b)(c,d)=(ac,ad+bc)$):

  1. start with $R=k[T]_{(T)}+k(T)$, with $k=$prime field contained in $K$;
  2. primary ideals of $R$ are $(0)+(0)$, $(0)+k(T)$, and $(T^n)+k(T)$ $(n>0)$;
  3. the ideal $I=(0)+k[T]_{(T)}$ is irreducible and not primary;
  4. $R$ is countable so it's a quotient of $k[X_1,X_2,...]$;
  5. extend scalars from $k[X_1,X_2,...]$ to $K[X_1,X_2,...]$

The original answer here was incorrect and I removed it. There is a different answer "using idealization construction" submitted with this question.

Another example using idealization construction (i.e. $(a,b)(c,d)=(ac,ad+bc)$):

  1. start with $R=k[T]_{(T)}+k(T)$, with $k=$prime field contained in $K$;
  2. primary ideals of $R$ are $(0)+(0)$, $(0)+k(T)$, and $(T^n)+k(T)$ $(n>0)$;
  3. the ideal $I=(0)+k[T]_{(T)}$ is irreducible and not primary;
  4. $R$ is countable so it's a quotient of $k[X_1,X_2,...]$;
  5. extend scalars from $k[X_1,X_2,...]$ to $K[X_1,X_2,...]$
deleted 760 characters in body
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David Lampert
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IThe original answer here was mistaken, this "better example" doesn't work: (X1-X32, X2-X43, X1-X55, X2-X67, X1-X711, X2-X813, ..., X3X4, X5X6, X7X8, ... )

In other terms this is: (X1/2Y1/3, X1/5Y1/7, ...) in the ring k[X,Y,X1/2,Y1/3,X1/5,Y1/7,...]

Which is the intersection:

(X1/2,X1/5Y1/7,..,) ∩ (Y1/3,X1/5Y1/7,...)

incorrect and I apologize for the confusionremoved it. I don't mind returning the bounty points There is a different answer "using idealization construction" submitted with this question.

I was mistaken, this "better example" doesn't work: (X1-X32, X2-X43, X1-X55, X2-X67, X1-X711, X2-X813, ..., X3X4, X5X6, X7X8, ... )

In other terms this is: (X1/2Y1/3, X1/5Y1/7, ...) in the ring k[X,Y,X1/2,Y1/3,X1/5,Y1/7,...]

Which is the intersection:

(X1/2,X1/5Y1/7,..,) ∩ (Y1/3,X1/5Y1/7,...)

I apologize for the confusion. I don't mind returning the bounty points.

The original answer here was incorrect and I removed it. There is a different answer "using idealization construction" submitted with this question.

deleted 105 characters in body
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David Lampert
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I was mistaken, this "better example" doesn't work: (X1-X32, X2-X43, X1-X55, X2-X67, X1-X711, X2-X813, ..., X3X4, X5X6, X7X8, ... )

In other terms this is: (X1/2Y1/3, X1/5Y1/7, ...) in the ring k[X,Y,X1/2,Y1/3,X1/5,Y1/7,...]

Which is the intersection:

(X1/2,X1/5Y1/7,..,) ∩ (Y1/3,X1/5Y1/7,...)

I'm undecided if the original answer ("I think this works") is correct, I'll try to report again later. II apologize for the confusion. I don't mind returning the bounty points.

I was mistaken, this "better example" doesn't work: (X1-X32, X2-X43, X1-X55, X2-X67, X1-X711, X2-X813, ..., X3X4, X5X6, X7X8, ... )

In other terms this is: (X1/2Y1/3, X1/5Y1/7, ...) in the ring k[X,Y,X1/2,Y1/3,X1/5,Y1/7,...]

Which is the intersection:

(X1/2,X1/5Y1/7,..,) ∩ (Y1/3,X1/5Y1/7,...)

I'm undecided if the original answer ("I think this works") is correct, I'll try to report again later. I apologize for the confusion. I don't mind returning the bounty points.

I was mistaken, this "better example" doesn't work: (X1-X32, X2-X43, X1-X55, X2-X67, X1-X711, X2-X813, ..., X3X4, X5X6, X7X8, ... )

In other terms this is: (X1/2Y1/3, X1/5Y1/7, ...) in the ring k[X,Y,X1/2,Y1/3,X1/5,Y1/7,...]

Which is the intersection:

(X1/2,X1/5Y1/7,..,) ∩ (Y1/3,X1/5Y1/7,...)

I apologize for the confusion. I don't mind returning the bounty points.

added 60 characters in body
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David Lampert
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The monomial construction doesn't work
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David Lampert
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added the word "example" in "the example ideal"
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David Lampert
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Bounty Ended with 50 reputation awarded by user26857
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David Lampert
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