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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?2012-04-25

1 Answers 1

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(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!2012-04-25
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    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?2012-04-25
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    @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.2012-04-25
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    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.2012-04-25
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    @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,2012-04-25
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    will you upload your answer please.2012-04-25
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    @zilexu For which part? And, have you tried solving that? Where are you stuck? :-)2012-04-25
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    I just totally dont understand the relation between subrings ideal and what exactly part 2 ask me to do.2012-04-25