Maximal quotients of semiprime PI-algebras

Rowen, Louis Halle

doi:10.1090/S0002-9947-1974-0347887-8

Maximal quotients of semiprime PI-algebras
HTML articles powered by AMS MathViewer

by Louis Halle Rowen

Trans. Amer. Math. Soc. 196 (1974), 127-135

DOI: https://doi.org/10.1090/S0002-9947-1974-0347887-8

PDF | Request permission

Abstract:

J. Fisher [3] initiated the study of maximal quotient rings of semiprime PI-rings by noting that the singular ideal of any semiprime Pi-ring R is 0; hence there is a von Neumann regular maximal quotient ring $Q(R)$ of R. In this paper we characterize $Q(R)$ in terms of essential ideals of C = cent R. This permits immediate reduction of many facets of $Q(R)$ to the commutative case, yielding some new results and some rapid proofs of known results. Direct product decompositions of $Q(R)$ are given, and $Q(R)$ turns out to have an involution when R has an involution.

References

S. A. Amitsur, On rings of quotients, Symposia Mathematica, Vol. VIII (Convegno sulle Algebre Associative, INDAM, Rome, 1970) Academic Press, London, 1972, pp. 149–164. MR 0332855
Efraim P. Armendariz and Stuart A. Steinberg, Regular self-injective rings with a polynomial identity, Trans. Amer. Math. Soc. 190 (1974), 417–425. MR 354763, DOI 10.1090/S0002-9947-1974-0354763-3
Joe W. Fisher, Structure of semiprime P.I. rings, Proc. Amer. Math. Soc. 39 (1973), 465–467. MR 320049, DOI 10.1090/S0002-9939-1973-0320049-0
Edward Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129–132. MR 302689, DOI 10.1016/0021-8693(72)90050-6
R. E. Johnson, Quotient rings of rings with zero singular ideal, Pacific J. Math. 11 (1961), 1385–1392. MR 143779, DOI 10.2140/pjm.1961.11.1385
Wallace S. Martindale III, On semiprime P. I. rings, Proc. Amer. Math. Soc. 40 (1973), 365–369. MR 318215, DOI 10.1090/S0002-9939-1973-0318215-3
Claudio Procesi, On a theorem of M. Artin, J. Algebra 22 (1972), 309–315. MR 302681, DOI 10.1016/0021-8693(72)90148-2
Louis Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219–223. MR 309996, DOI 10.1090/S0002-9904-1973-13162-3
Louis Halle Rowen, A subdirect decomposition of semiprime rings and its application to maximal quotient rings, Proc. Amer. Math. Soc. 46 (1974), 176–180. MR 349728, DOI 10.1090/S0002-9939-1974-0349728-7