Classical quotient rings of PWD's

Author:
Robert Gordon

Journal:
Proc. Amer. Math. Soc. **36** (1972), 39-46

MSC:
Primary 16A08

MathSciNet review:
0309983

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Abstract: Piecewise domains which are right orders in semiprimary rings are characterized. An example is given showing the result obtained is ``best possible". A further example is obtained of a prime right Goldie ring possessing a regular element which becomes a left zero divisor in some prime overring. This example leads to the construction of a PWD *R* not satisfying the regularity condition, but for which is right Goldie.

**[1]**Paul M. Cohn,*On the free product of associative rings. II. The case of (skew) fields*, Math. Z.**73**(1960), 433–456. MR**0113916****[2]**Robert Gordon,*Rings faithfully represented on their left socle*, J. Algebra**7**(1967), 303–342. MR**0218396****[3]**Robert Gordon,*Semi-prime right Goldie rings which are direct sums of uniform right ideals*, Bull. London Math. Soc.**3**(1971), 277–282. MR**0294403****[4]**Robert Gordon and L. W. Small,*Piecewise domains*, J. Algebra**23**(1972), 553–564. MR**0309984****[5]**I. N. Herstein and Lance W. Small,*Regular elements in 𝑃.𝐼.-rings*, Pacific J. Math.**36**(1971), 327–330. MR**0281751****[6]**Arun Vinayak Jategaonkar,*Ore domains and free algebras*, Bull. London Math. Soc.**1**(1969), 45–46. MR**0238881****[7]**Lawrence Levy,*Torsion-free and divisible modules over non-integral-domains*, Canad. J. Math.**15**(1963), 132–151. MR**0142586****[8]**J. C. Shepherdson,*Inverses and zero divisors in matrix rings*, Proc. London Math. Soc. (3)**1**(1951), 71–85. MR**0041831****[9]**Lance W. Small,*Orders in Artinian rings. II*, J. Algebra**9**(1968), 266–273. MR**0230755**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0309983-4

Keywords:
Piecewise domain,
essentiality condition,
exhaustive set,
order,
classical quotient ring,
Goldie ring,
principal right ideals projective,
torsion-free,
left regularity condition,
Ore domain,
linear equations

Article copyright:
© Copyright 1972
American Mathematical Society