Maximal quotients of semiprime PIalgebras
Author:
Louis Halle Rowen
Journal:
Trans. Amer. Math. Soc. 196 (1974), 127135
MSC:
Primary 16A38
MathSciNet review:
0347887
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Abstract: J. Fisher [3] initiated the study of maximal quotient rings of semiprime PIrings by noting that the singular ideal of any semiprime Piring R is 0; hence there is a von Neumann regular maximal quotient ring of R. In this paper we characterize in terms of essential ideals of C = cent R. This permits immediate reduction of many facets of to the commutative case, yielding some new results and some rapid proofs of known results. Direct product decompositions of are given, and turns out to have an involution when R has an involution.
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S. A. Amitsur, On rings of quotients, Istituto Nazionale di Alta Matematica, Symposia Matematica, vol. VIII, 1972. MR 0332855 (48:11180)
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E. Armendariz and S. Steinberg, Regular selfinjective rings with a polynomial identity, Trans. Amer. Math. Soc. 190 (1974), 417425. MR 0354763 (50:7240)
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J. W. Fisher, Structure of semiprime P.I. rings. I, Proc. Amer. Math. Soc. 39 (1973), 465467. MR 0320049 (47:8590)
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E. Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129132. MR 46 #1833. MR 0302689 (46:1833)
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R. E. Johnson, Quotient rings of rings with zero singular ideal, Pacific J. Math. 11 (1961), 13851392. MR 26 #1331. MR 0143779 (26:1331)
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W. S. Martindale III, On semiprime P.I. rings, Proc. Amer. Math. Soc. 40 (1973), 365369. MR 0318215 (47:6762)
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C. Procesi, On a theorem of M. Artin, J. Algebra 22 (1972), 309315. MR 46 #1825. MR 0302681 (46:1825)
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L. H. Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219223. MR 0309996 (46:9099)
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, A subdirect decomposition of semiprime rings and its application to maximal quotient rings, Proc. Amer. Math. Soc. (to appear). MR 0349728 (50:2221)
 [1]
 S. A. Amitsur, On rings of quotients, Istituto Nazionale di Alta Matematica, Symposia Matematica, vol. VIII, 1972. MR 0332855 (48:11180)
 [2]
 E. Armendariz and S. Steinberg, Regular selfinjective rings with a polynomial identity, Trans. Amer. Math. Soc. 190 (1974), 417425. MR 0354763 (50:7240)
 [3]
 J. W. Fisher, Structure of semiprime P.I. rings. I, Proc. Amer. Math. Soc. 39 (1973), 465467. MR 0320049 (47:8590)
 [4]
 E. Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129132. MR 46 #1833. MR 0302689 (46:1833)
 [5]
 R. E. Johnson, Quotient rings of rings with zero singular ideal, Pacific J. Math. 11 (1961), 13851392. MR 26 #1331. MR 0143779 (26:1331)
 [6]
 W. S. Martindale III, On semiprime P.I. rings, Proc. Amer. Math. Soc. 40 (1973), 365369. MR 0318215 (47:6762)
 [7]
 C. Procesi, On a theorem of M. Artin, J. Algebra 22 (1972), 309315. MR 46 #1825. MR 0302681 (46:1825)
 [8]
 L. H. Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219223. MR 0309996 (46:9099)
 [9]
 , A subdirect decomposition of semiprime rings and its application to maximal quotient rings, Proc. Amer. Math. Soc. (to appear). MR 0349728 (50:2221)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403478878
PII:
S 00029947(1974)03478878
Keywords:
Essential,
identity,
injective hull,
involution,
maximal quotient algebra,
PIalgebra,
semiprime,
singular ideal
Article copyright:
© Copyright 1974
American Mathematical Society
