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Establish, whether or not following subsets of given rings are subrings

  1. All polynomials $f(x)$ with $f(0)=9$ in $\mathbb{Z}[x]$.

These are past paper questions, I have no clue what $\mathbb{Z}[x]$ is, can anyone give me some help please. There are also two more questions:

Establish, whether or not following subsets of given rings are ideals:

  1. All integers divisible by $5$ in $\mathbb{Q}$ ($\mathbb{Q}$ is the field of rational numbers).

  2. All polynomials in $\mathbb{Z}[x]$ with coefficients divisible by $5$ in $\mathbb{Z}[x]$.

Thank you so much

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$\mathbb Z[X]$ is the ring of polynomials with integer coefficients. If you haven't seen this notation before (or its general form $R[X]$ for an arbitrary ring $R$), are you sure you should be trying to solve that problem set in the first place? – Henning Makholm Apr 25 '12 at 0:06

(Answer will be updated if OP or others asks for more details; Please leave me a comment and I shall elaborate.)

Part 1

The notation, $\Bbb{Z}[X]$, stands for the ring of polynomials with integer coefficients.

Part 2

To verify that a set $I$ is an ideal of a commutative ring $R$, you will have to verify that:

  1. $I$ is an additive subgroup of $R$.
  2. For any $r \in R$ and $i \in I$, $ri \in R$. (I call this by a name: Extended closure under multiplication.)

I'll leave it at this for now. If you have dificulty, we can fill in the details together. But, try to take the problem from here for now.

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Look who updated their avatar! – The Chaz 2.0 Apr 25 '12 at 1:13
Why are you doing so much on Part 1 when all that OP asked for was the meaning of one piece of notation? Why are you robbing OP of the joy of figuring things out? – Gerry Myerson Apr 25 '12 at 4:30
@GerryMyerson I intended not to do exactly that--but that you tell me, I feel I have done just the opposite. I thought I just wrote down the definitions. I am sorry. What part of my answer do you want me to get rid of? Will oblige. – user21436 Apr 25 '12 at 4:32
OP didn't ask for the definitions, just for the meaning of one piece of notation. But re-reading my comment I think it was worded too harshly. Do as you see fit, I won't complain whatever you decide. – Gerry Myerson Apr 25 '12 at 6:10
@GerryMyerson I have edited it out. But, for part-2, I am clueless if OP wrote those questions to give context about $\Bbb{Z}[X]$ or really wants help on them. So, I have left that as such. Regards, – user21436 Apr 25 '12 at 7:26

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