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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The normal closure of the coproduct of rings over a division ring


Author: Wallace S. Martindale
Journal: Trans. Amer. Math. Soc. 293 (1986), 303-317
MSC: Primary 16A06; Secondary 16A03, 16A08
MathSciNet review: 814924
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Abstract: Let $ R = {R_1}\coprod {R_2}$ be the coproduct of $ \Delta$-rings $ {R_1}$ and $ {R_2}$ with 1 over a division ring $ \Delta ,\qquad {R_1} \ne \Delta ,\qquad {R_2} \ne \Delta $, with at least one of the dimensions $ {({R_i}:\Delta )_r},\,{({R_i}:\Delta )_l},\,i = 1,\,2$, greater than 2. If $ {R_1}$ and $ {R_2}$ are weakly $ 1$-finite (i.e., one-sided inverses are two-sided) then it is shown that every $ X$-inner automorphism of $ R$ (in the sense of Kharchenko) is inner, unless $ {R_1},\,{R_2}$ satisfy one of the following conditions: (I) each $ {R_i}$ is primary (i.e., $ {R_i} = \Delta + T,\,{T^2} = 0$), (II) one $ {R_i}$ is primary and the other is $ 2$-dimensional, (III) char. $ \Delta = 2$, one $ {R_i}$ is not a domain, and one $ {R_i}$ is $ 2$-dimensional. This generalizes a recent joint result with Lichtman (where each $ {R_i}$ was a domain) and an earlier joint result with Montgomery (where each $ {R_i}$ was a domain and $ \Delta$ was a field).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0814924-1
PII: S 0002-9947(1986)0814924-1
Article copyright: © Copyright 1986 American Mathematical Society