general elements of $\frac{k[x,y,z]}{I}$ and basis

Question

for

1. $\frac{k[x,y,z]}{y-z,xz-z}$

2. $\frac{k[x,y,z]}{y-z,x-1}$

3. $\frac{k[x,y,z]}{y-z,z}$

Identify a basis for $k[x,y,z]$ and List general elements of $k[x,y,z]/I$

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for (a) $$\frac{k[x,y,z]}{y-z,xz-z} $$ basis is $$1,x^i ,z^j ;i \geq 1 , j \geq 1 $$

and elements are finitely generated

$$a + \sum_{j \geq 1} b_i x^j + \sum_{j \geq 1 } c_j z^j$$

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These are examples from lecture but had a trouble fowlling it.

I thouhgt that the base would be $$ 1,x,z$$

but the highest power would be $x,z$ of $1$

Answer 1

From the computation in the middle of your post, it sounds like you're asking for a basis as a $k$-module. To explain that computation more thoroughly, note that every time $y$ appears in an element, you may replace it by $z$ as $y-z=0$ and similarly, every time $xz$ appears in an element, you may replace it by $z$.

So if you start with $\sum_{i,j,k\geq 0} a_{ijk}x^iy^jz^k$, you may turn this in to $\sum_{i,j,k\geq 0} a_{ijk}x^iz^{j+k}$ and then $x^iz^{j+k}$ may be replaced with $z^{j+k}$ unless $j+k=0$. This shows that the elements $1,x^p,z^q$ for $p,q>1$ form a $k$-basis of this module.

This procedure may be repeated for each of the other modules listed. You should be able to systematically apply rewriting rules to elements in each of these to get the following bases:

2:

$y^j$ for $j\geq0$

3:

$x^j$ for $j\geq 0$