# 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?