Abstract algebra

Ab­stract alge­bra is the study of alge­braic struc­tures, in­clud­ing groups, rings, fields, mod­ules, vec­tor spaces, lat­tices, ar­ith­metics, and alge­bras.

The main idiom of ab­stract alge­bra is ab­stract­ing away from the ob­jects: Ab­stract alge­bra con­cerns it­self with the ma­nipu­la­tion of ob­jects, by fo­cus­ing not on the ob­jects them­selves but on the re­la­tion­ships be­tween them.

If you find any col­lec­tion of ob­jects that are re­lated to each other in a man­ner that fol­lows the laws of some alge­braic struc­ture, then those re­la­tion­ships are gov­erned by the cor­re­spond­ing the­o­rems, re­gard­less of what the ob­jects are. An ab­stract alge­braist does not ask “what are num­bers, re­ally?”; rather, they say “I see that the op­er­a­tions of ‘adding ap­ples to the table’ and ‘re­mov­ing ap­ples from the table’ fol­low the laws of num­bers (in a limited do­main), thus, the­o­rems about num­bers can tell what to ex­pect as I add or re­move ap­ples (in that limited do­main).”

For a map of alge­braic struc­tures and how they re­late to each other, see the tree of alge­braic struc­tures.

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  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.