## Simple Algebra does not imply Semisimple Algebra

The terminology “semisimple” algebra suggests a generalization of simple algebras, but in fact not all simple algebras are semisimple! (Exercises 1 & 5 in Richard Pierce’s book contain examples)

A simple module is a semisimple module is true though.

Proposition: For a simple algebra A, the following conditions are equivalent:

(i) A is semisimple;

(ii) A is right Artinian;

(iii) A has a minimal right ideal.

Thus to find a algebra that is simple but not semisimple, one can look for an example that is not right Artinian.
{"ft_lang_label":"__label__en","ft_lang_prob":0.81423426,"math_prob":0.9839875,"size":523,"snap":"2019-51-2020-05","text_gpt3_token_len":131,"char_repetition_ratio":0.18111753,"word_repetition_ratio":0.0,"special_character_ratio":0.21414913,"punctuation_ratio":0.11111111,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9684237,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-27T06:00:16Z\",\"WARC-Record-ID\":\"<urn:uuid:63343134-49b5-47f7-8b9b-7ebb9cc7a87f>\",\"Content-Length\":\"86915\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cea54c57-53a2-4fbc-ace9-d196a2bbd9bd>\",\"WARC-Concurrent-To\":\"<urn:uuid:9fb48da3-45df-424d-9c23-056553ba661c>\",\"WARC-IP-Address\":\"192.0.78.24\",\"WARC-Target-URI\":\"https://mathtuition88.com/2016/02/27/simple-algebra-does-not-imply-semisimple-algebra/\",\"WARC-Payload-Digest\":\"sha1:VOER3TTQTXXIU73Y7VYX5TQ7BQIHMO2X\",\"WARC-Block-Digest\":\"sha1:MD4MRVDVZEXE2Y22IHOREEJMAZ3BGYZD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579251694908.82_warc_CC-MAIN-20200127051112-20200127081112-00513.warc.gz\"}"}