I'm just learning basic ring theory and had a question about the definition of a principal ideal. For a commutative ring $R$ with unity, Fraleigh defines the principal ideal generated by $a\in R$ as $\langle a\rangle = \{ra:r\in R\}$. My question is, why do we need to assume that $R$ has a unity? It seems like $\langle a\rangle$ fits the criterion for an ideal without it. Is this the standard definition? If so, do we include the unity condition so that we can say that $a\in \langle a\rangle$? Also, if we require that every ring have a unity, wouldn't we have that the entire ring is a principal ideal of itself, since $R=\langle1\rangle$?
• If you like to write ideals with angly-brackets, use \langle and \rangle, not < and >, so that you get $\langle a\rangle$ instead of the immensly uglier $<a>$ :-) – Mariano Suárez-Álvarez Apr 13 '14 at 5:51
• I am not familiar with non-unital ring theory, but the fact that $a\in Ra$ requires $1\in R$ seems relevant. There is another definition of principal ideal as that generated by a single element, where we define the ideal generated by a set to be the smallest ideal containing the set (so it is the intersection of all such). On this definition, $Ra$ would be different from $\langle a\rangle$ (the smallest ideal containing $a$) if $R$ doesn't have a $1$. In $R=2\Bbb Z$ for instance, $2R=4\Bbb Z$ but $\langle2\rangle=2\Bbb Z$. Also yes, $R=\langle1\rangle$ in any unital ring (why do you ask?). – anon Apr 13 '14 at 5:57
{"ft_lang_label":"__label__en","ft_lang_prob":0.90098804,"math_prob":0.9967754,"size":837,"snap":"2019-26-2019-30","text_gpt3_token_len":237,"char_repetition_ratio":0.1452581,"word_repetition_ratio":0.0,"special_character_ratio":0.27359617,"punctuation_ratio":0.105882354,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99987125,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-23T02:35:16Z\",\"WARC-Record-ID\":\"<urn:uuid:c8902ceb-e054-450c-9fd0-2875eb49c33e>\",\"Content-Length\":\"140336\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d6a323c5-9524-41ed-a68d-1f90646364df>\",\"WARC-Concurrent-To\":\"<urn:uuid:9f4e6e24-671d-493c-84f8-fbc9a7fcbfe7>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/751478/question-about-principal-ideals\",\"WARC-Payload-Digest\":\"sha1:MKDAVZZS2DEZHI5XCQQ6D7BGKM6O63OL\",\"WARC-Block-Digest\":\"sha1:WQBTWCPBGW72WP2J4YN2C5IMXKWC5O72\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195528687.63_warc_CC-MAIN-20190723022935-20190723044935-00448.warc.gz\"}"}