My research is in analysis, but it moved to the area that requires algebraic topology. I have some working knowledge in that area, but I always feel that I am on a shaky ground and I need to go back and study algebraic topology again. However, that would also require refreshing my knowledge in algebra. My question is:



> 
> What is a good reference from which I could learn algebra necessary
>  for studying algebraic topology at the level of Hatcher's Algebraic Topology plus Eilenberg-Steenrod's axioms (not included in Hatcher's book) plus spectral sequences (in unpublished notes of Hatcher).
> 
> 
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I would love to find an elementary reference that would cover all necessary algebraic tools (including homological algebra) on no more than 100$\pm\varepsilon$ pages. 



Algebra for algebraic topology
If you need a concise but **very** clear book which covers a lot of Algebraic Topology and just the necessary algebra (spectral sequences as well) I think that *Differential Forms in Algebraic Topology*- Bott & Tu is the book you are looking for. 


Edit: It seems that Bredon-Topology and Geometry is closer to that you are looking for.


