Skip to main content
×
×
Home

A note on extensions of Baer and P. P. -rings

  • Efraim P. Armendariz (a1)
Extract

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jøndrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A note on extensions of Baer and P. P. -rings
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      A note on extensions of Baer and P. P. -rings
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      A note on extensions of Baer and P. P. -rings
      Available formats
      ×
Copyright
References
Hide All
[1]Bergman, G., ‘Hereditary commutative rings and centres of hereditary rings’, Proc. London Math. Soc. 23 (1971), 214236.
[2]Endo, S., ‘A Note on pp. rings’, Nagoya Math. J. 17 (1960), 167170.
[3]Evans, M., ‘On commutative P. P. rings’, Pacific J. Math. 41 (1924), 687697.
[4]Jacobson, N., Structure of Rings, (Amer. Math. Soc. Colloq. Publ. 37, Providence, R. I. (1964).)
[5]Jøndrup, S., ‘p. p. Rings and finitely generated fiat ideals’, Proc. Amer. Math. Soc. 28 (1971), 431435.
[6]Kaplansky, I., Rings of Operators (W. A. Benjamin, New York (1968)).
[7]Speed, T., ‘A note on commutative Baer rings’, J. Austral. Math. Soc. 14 (1972), 257263.
[8]Renault, G., ‘Anneaux reduits non commutatifs’, J. Math. Pures et Appl. 46 (1967), 203214.
[9]Rowen, L., ‘Some results on the center of a ring with polynomial identity’, Bull. Amer. Math. Soc. 79 (1973), 219223.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed