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Prime Ideals in Regular Self-Injective Rings

Published online by Cambridge University Press:  20 November 2018

K. R. Goodearl
K. R. Goodearl*
Affiliation:
University of Chicago, Chicago, Illinois; University of Utah, Salt Lake City, Utah
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Although the notion of the maximal quotient ring of a nonsingular ring has been around for some time, not much is known about its structure in general beyond the important theorems of Johnson and Utumi [4; 11] that it is von Neumann regular and self-injective. The purpose of this paper is to study the structure of such a regular, self-injective ring R by looking at its prime ideals. Initially, we show that the primes of R separate into two types, called ‘'essential” and ‘“closed”, and that for any prime P, the two-sided ideals in the ring R/P are linearly ordered.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 4 , August 1973 , pp. 829 - 839
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Bumby, R. T., Modules which are isomorphic to submodules of each other, Arch. Math. 16 (1965), 184–185.Google Scholar
2. Faith, C., Lectures on infective modules and quotient rings, Springer-Verlag Lecture Notes, No. 49.Google Scholar
3. Jacobson, N., Lectures in abstract algebra, Vol. II (Van Nostrand, Princeton, 1953).Google Scholar
4. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891–895.Google Scholar
5. Kaplansky, I., Algebraic and analytic aspects of operator algebras, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, No. 1.Google Scholar
6. Lambek, J., On the structure of semi-prime rings and their rings of quotients, Can. J. Math. 13 (1961), 392–417.Google Scholar
7. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, 1966).Google Scholar
8. Osofsky, B. L., Cyclic infective modules of full linear rings, Proc. Amer. Math. Soc. 17 (1966), 247–253.Google Scholar
9. Osofsky, B. L., Homological dimension and cardinality, Trans. Amer. Math. Soc. 151 (1970), 641–649.Google Scholar
10. Sandomierski, F. L., Nonsingular rings, Proc. Amer. Math. Soc. 19 (1968), 225–230.Google Scholar
11. Utumi, Y., On quotient rings, Osaka J. Math. 8 (1956), 1–18.Google Scholar