Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $k$ be a field and consider the ideal $I=(x,y) \subset k[x,y]$. Am I correct in saying that $(x,y)/(x,y)^{2}$ is generated as a $k$-vector space by the class of $x$ and $y$?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Yes. We have that $$(x,y)=\{f\in k[x,y]\mid f \text{ has no terms of degree}\leq0\}$$

and that $$(x,y)^2=(x^2,xy,y^2)=\{f\in k[x,y]\mid f \text{ has no terms of degree}\leq1\}$$

so that any $f\in(x,y)$ is equivalent modulo $(x,y)^2$ to one having terms only in degree 1, i.e. an element of $(x,y)$ of the form $ax+by$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.