Let k a field.

Let $ A = k+ x^2 k[x] $,

Show that A is integral domain and finite type.

An element of $A$ is $\alpha + x^2. f(x) $ where$ f\in k[x] $

I know that A is a subalgebra of k[x], Any hint for integral domain


Since $A\subset k[x]$ and since $k[x]$ is an integral domain, we can see that for all $a,b\in A$, $(ab = 0$ in $A) \Rightarrow (ab = 0$ in $k[x]) \Rightarrow (a=0$ or $b=0)$.

  • $\begingroup$ An element of $A$ is $\alpha + x^2. f(x) $ when $ f\in k[x] $ $\endgroup$ – Ali NoumSali Traore Aug 6 '18 at 20:06
  • $\begingroup$ Yes and such an element is a polynomial in $k[x]$. $\endgroup$ – paf Aug 6 '18 at 20:07
  • $\begingroup$ Isn't $(x^2)$ a prime ideal of $A$? Then the quotient $A/(x^2)$ is...? $\endgroup$ – misogrumpy Aug 6 '18 at 20:36

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