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$\begingroup$

I was wondering what the notation $R[a]$ really stands for, if $a\in K$, where $K$ is a ring and $R$ is a subring of $K$.

In my book they define $\mathbb{Z}[\sqrt2]=\{a+b\sqrt2|a,b \in \mathbb{Z}\}$.

So, my guess is that $R[a] = \{P(a)\mid P \in R[X]\}$. Since for $\mathbb{Z}[\sqrt2]$ this is the case, or is this just a coincidence?

  • 1
    That looks fine to me.2012-02-04
  • 3
    It is a simple *ring adjunction*, see http://math.stackexchange.com/questions/15453/meaning-of-mathbbrx2012-02-04
  • 3
    See also [this answer](http://math.stackexchange.com/a/86770/14366).2012-02-04

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