Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-24T23:12:14.839Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  09 April 2009

Efraim P. Armendariz
Affiliation:
University of TexasAustin, Texas 78127, U. S. A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1974

References

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