In working with two ideals in $k[x_1,\ldots,x_n]$ where $k$ is a field I know that the all elements in both have degree greater than or equal to $x$ and that every element of degree $x$ in ideal $A$ is in ideal $B$ and every element of degree $x$ in $B$ is in $A$. Can I somehow conclude from this that $A=B$?
This would be easy if I knew that both $A$ and $B$ were generated only by elements of degree $x$, and I know that this is true of $A$, but I do not know that it is true for $B$.