Strongly prime rings

Authors:
David Handelman and John Lawrence

Journal:
Trans. Amer. Math. Soc. **211** (1975), 209-223

MSC:
Primary 16A12

MathSciNet review:
0387332

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Abstract: A ring *R* is (right) strongly prime (SP) if every nonzero twosided ideal contains a finite set whose right annihilator is zero. Examples are domains, prime Goldie rings and simple rings; however, this notion is asymmetric and a right but not left SP ring is exhibited. All SP rings are prime, and every prime ring may be embedded in an SP ring. SP rings are nonsingular, and a regular SP ring is simple; since faithful rings of quotients of SP rings are SP, the complete ring of quotients of an SP ring is simple.

All SP rings are coefficient rings for some primitive group ring (a generalization of a result proved for domains by Formanek), and this was the initial motivation for their study. If the group ring *RG* is SP, then *R* is SP and *G* contains no nontrivial locally finite normal subgroups.

Coincidentally, SP rings coincide with the ATF rings of Rubin, and so every SP ring has a unique maximal proper torsion theory, and (0) and *R* are the only torsion ideals.()

A list of questions is appended.

**[1]**Ian G. Connell,*On the group ring*, Canad. J. Math.**15**(1963), 650–685. MR**0153705****[2]**Carl Faith,*Lectures on injective modules and quotient rings*, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR**0227206****[3]**Edward Formanek,*Group rings of free products are primitive*, J. Algebra**26**(1973), 508–511. MR**0321964****[4]**A. W. Goldie,*Torsion-free modules and rings*, J. Algebra**1**(1964), 268–287. MR**0164991****[5]**Oscar Goldman,*Rings and modules of quotients*, J. Algebra**13**(1969), 10–47. MR**0245608****[6]**K. R. Goodearl,*Prime ideals in regular self-injective rings*, Canad. J. Math.**25**(1973), 829–839. MR**0325690****[7]**I. N. Herstein,*Noncommutative rings*, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR**0227205****[8]**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106****[9]**Irving Kaplansky,*Algebraic and analytic aspects of operator algebras*, American Mathematical Society, Providence, R.I., 1970. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 1. MR**0312283****[10]**Joachim Lambek,*Lectures on rings and modules*, With an appendix by Ian G. Connell, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. MR**0206032****[11]**Joachim Lambek,*Torsion theories, additive semantics, and rings of quotients*, With an appendix by H. H. Storrer on torsion theories and dominant dimensions. Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. MR**0284459****[12]**J. Lawrence,*Primitivr group rings*, Master's Thesis, McGill Univ., Montreal, Canada, 1973.**[13]**S. Page,*Properties of quotient rings*, Canad. J. Math.**24**(1972), 1122–1128. MR**0311700****[14]**Robert A. Rubin,*Absolutely torsion-free rings*, Pacific J. Math.**46**(1973), 503–514. MR**0364326****[15]**Bo Stenström,*Rings and modules of quotients*, Lecture Notes in Mathematics, Vol. 237, Springer-Verlag, Berlin-New York, 1971. MR**0325663****[16]**J. Viola-Prioli,*On absolutely torsion-free rings and kernel functors*, Ph.D. Thesis, Rutgers University, New Brunswick, N. J., 1973.**[17]**P. M. Cohn,*On the free product of associative rings*, Math. Z.**71**(1959), 380–398. MR**0106918****[18]**Paul M. Cohn,*On the free product of associative rings. II. The case of (skew) fields*, Math. Z.**73**(1960), 433–456. MR**0113916**

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0387332-0

Article copyright:
© Copyright 1975
American Mathematical Society