Knowing Sandor, I will heartily second what he said. On second mention, I will say that Mumford's red book on algebraic geometry begins with 5-10 pages (depending on your edition) called "some algebra". This consists of the following subset of Sandor's list: noether's normalization lemma, and cohen - seidenberg's going up lemma, plus corollaries such as the "weak" nullstellensatz. Hence this seems to be a minimum of algebra to know.

I remark that this is also in sync with the post by Dmitry that not that much is necessary.