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nLab > Latest Changes: semisimple ring

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  1. starting article on semisimple rings

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorJohn Baez
    • CommentTimeJun 17th 2023

    Added two equivalent definitions of “semisimple ring”; changed “semiprimitive” to “with vanishing Jacobson radical” since this is a bit more familiar to many people.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorJohn Baez
    • CommentTimeJun 17th 2023

    Added more examples of semisimple rings, and also examples of simple rings that are not semisimple, with an explanation of why this occurs.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeJul 1st 2023

    Two more characterizations of semisimple rings:

    • Every left RR-module is projective

    • Every right RR-module is projective.

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorJoseBrox
    • CommentTimeSep 27th 2023
    The current page says "The ring of linear transformations of an infinite-dimensional vector space over field is also simple but not Artinian and thus not semisimple". This is not true: this ring is prime with socle but not simple, since it does not coincide with its socle, which is the subring (without identity) of linear transformations of finite rank. Some modification is needed.
    • CommentRowNumber6.
    • CommentAuthorJohn Baez
    • CommentTimeOct 3rd 2023

    I don’t follow your logic, since I don’t know much about socles. But I’ll just delete that (non)example.

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