Fields of finite order are well classified, and classification of groups of finite order has taken some depth in research. Why classification of finite rings and modules is not well studied in research?
Rings and modules of finite order
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abstract-algebra
ring-theory
finite-rings
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1Finite rings are studied; you just need to know where to look. However, most basic algebra texts do not study finite rings deeply simply due to lack of space. – 2011-06-09
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1why a downvote, i think this question is a decent one – 2011-06-09
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1The downvote is not mine, but the question has two faulty premises: (i) there is no classification of groups of finite order (only finite simple groups) and (ii) finite rings *are* very well-studied in research. If the OP had done a quick internet search, s/he would find many papers on the latter. So the question does not look so well thought out. – 2011-06-09