anarchy, the presumed objective of anarchism, means - quite literally - something very close to "no rulers". so saying "rulers and governments could still exist" is not consistent with any concept of anarchy that i relate to.
i would say anarchism could be considered a political ideology, even though those with a more post/anti-left perspective would probably consider themselves anti-political as well as anti-ideology. as rs666 alludes to below, ideology is kind of a loaded word. the differences between ideology and dogma can be quite subtle, but they do exist. after some interesting recent discussion (irl) about exactly that, i want to try to be more clear when using those terms.
"for a critical examination of how Politics has become a negative term, take a brief look at the work of Michel Foucault"
what does this mean???