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I have the following exercise: I have a list of subsets of $\mathbb{Z}[X]$ and have to say which of these are subrings and which are ideals; and if it is an ideal I have to specify a generator for the set.

What is not clear to me is, how the "specify a generator for the set" is meant, since I could very well take the subset itself and it will generate itself obviously (for example, two of those subsets are set set of all polynomials with even coefficients and the set of all polynomials and the set of all polynomials whose coefficients at the terms of degree $0$ and $1$ vanish).

Is there an implicit assumption one makes, when one says that a generator is to be specified ? For example, is this generator maybe supposed to be a minimal generator (w.r.t to inclusion) ?

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    I would normally take "generator" to be singular - find a single element which generates all the others. This is not possible for all ideals of $\mathbb{Z}[X]$, though, so perhaps I'm also misunderstanding the question.2012-10-23
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    @JasonDeVito Yes, that is exactly what is unclear to me too!2012-10-23
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    If you want to know what the person who gave you this exercise meant, since it's either not using standard terminology or there's a typo, you're going to have to ask the person who gave you the exercise.2012-10-23
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    @NoahSnyder Ok if there's no general consensus (which I thought perhaps existed, but maybe I didn't knew it) exists how to interpret that phrase, this question ought to be deleted.2012-10-23
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    I'd suggest to add a tag "soft-question" (i.e. without a definitive answer), but not deleting the question right away.2013-08-18

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To give a few examples:

(1) $\langle x\rangle = x\mathbb{Z}[x]$. Here you simply have all the polynomials in $\mathbb{Z}[x]$ that are a multiple of $x$, i.e. that are divisible by $x$. Hence $x\mathbb{Z}[x] = \{\sum_{i=1}^{n} a_ix^i\lvert $finite sums$\}$.

(2) $\langle 2x^3+3\rangle = (2x^3+3)\mathbb{Z}[x]$

These are two ideals of $\mathbb{Z}[x]$. A generator of the first ideal is $x$. A generator of the second ideal is $(2x^3 + 3)$.

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    Yes, I know what a generator is. But, as JasonDeVito remarked, how about ideals in $\mathbb{Z}[X]$ that aren't generated by a single element ? Does the phrase "specify a generator" perhaps mean "specify a set containing a single element that generates the ideal" ? The exact meaning of "specify a generator" is what I want to know.2012-10-23