Let $k$ be a field. Let $q=(x,y^2)$ be an ideal of $k[x,y]$.
What exactly does the notation $q=(x,y^2)$ mean, i.e. what kind of elements does $q$ contain? Is it the set of all elements $\alpha x + \beta y^2, \, \, \alpha, \beta \in k$?
The ideal $(x,y^2)$ consists of all elements of the form $\alpha(x,y) x + \beta(x,y) y^2$ for $\alpha, \beta \in k[x,y]$. More generally, if $R$ is a commutative ring and $f_1, \dots, f_k \in R$, then $(f_1, \dots, f_k)$ consists of all elements of the form $\alpha_1 f_1 + \cdots \alpha_k f_k$ with $\alpha_i \in R$.