ag.algebraic geometry – Problem on $U_x$ the ideal of the closed set X

I’m working on the following question from Shafarevich’s Basic Algebraic Geometry I and want to check that I understand the ideal ${mathfrak U}_X$ of the closed set $X$.

The set $Xsubset {mathbb A}^2$ is defined by the equation $f:x^2+y^2=1$ and $g:x=1$. Find the ideal ${mathfrak U}_X$. Is it true that ${mathfrak U}_X = (f,g)$?

The ideal ${mathfrak U}_X$ consists of polynomials that are $0$ on all of $X$. Then these polynomials need to be divisible by $x-1$ and by $x^2+y^2-1$, so I think ${mathfrak U}_X$ is the ideal $(fg)$, not the ideal $(f,g)$. Is this correct? Am I correctly understanding how $X$ is being defined in this problem?