The normal closure of the coproduct of rings over a division ring
Author:
Wallace S. Martindale
Journal:
Trans. Amer. Math. Soc. 293 (1986), 303317
MSC:
Primary 16A06; Secondary 16A03, 16A08
MathSciNet review:
814924
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be the coproduct of rings and with 1 over a division ring , with at least one of the dimensions , greater than 2. If and are weakly finite (i.e., onesided inverses are twosided) then it is shown that every inner automorphism of (in the sense of Kharchenko) is inner, unless satisfy one of the following conditions: (I) each is primary (i.e., ), (II) one is primary and the other is dimensional, (III) char. , one is not a domain, and one is dimensional. This generalizes a recent joint result with Lichtman (where each was a domain) and an earlier joint result with Montgomery (where each was a domain and was a field).
 [1]
George
M. Bergman, Modules over coproducts of
rings, Trans. Amer. Math. Soc. 200 (1974), 1–32. MR 0357502
(50 #9970), http://dx.doi.org/10.1090/S00029947197403575025
 [2]
P.
M. Cohn, On the free product of associative rings, Math. Z.
71 (1959), 380–398. MR 0106918
(21 #5648)
 [3]
Paul
M. Cohn, On the free product of associative rings. II. The case of
(skew) fields, Math. Z. 73 (1960), 433–456. MR 0113916
(22 #4747)
 [4]
P.
M. Cohn, On the free product of associative rings. III, J.
Algebra 8 (1968), 376–383. MR 0222118
(36 #5170)
 [5]
Alexander
I. Lichtman and Wallace
S. Martindale III, The extended center of the coproduct of rings
over a division ring, Comm. Algebra 12 (1984),
no. 1718, 2067–2080. MR 747218
(85j:16021), http://dx.doi.org/10.1080/00927878408823098
 [6]
Alexander
I. Lichtman and Wallace
S. Martindale III, The normal closure of the coproduct of domains
over a division ring, Comm. Algebra 13 (1985),
no. 7, 1643–1664. MR 790402
(87b:16003), http://dx.doi.org/10.1080/00927878508823243
 [7]
Wallace
S. Martindale III, The extended center of coproducts, Canad.
Math. Bull. 25 (1982), no. 2, 245–248. MR 663623
(83h:16005), http://dx.doi.org/10.4153/CMB1982036x
 [8]
Wallace
S. Martindale III and Susan
Montgomery, The normal closure of coproducts of domains, J.
Algebra 82 (1983), no. 1, 1–17. MR 701033
(84k:16003), http://dx.doi.org/10.1016/00218693(83)901709
 [9]
Susan
Montgomery, Automorphism groups of rings with no nilpotent
elements, J. Algebra 60 (1979), no. 1,
238–248. MR
549109 (81j:16040), http://dx.doi.org/10.1016/00218693(79)901194
 [1]
 G. Bergman, Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 132. MR 0357502 (50:9970)
 [2]
 P. M. Cohn, On the free product of associative rings, Math. Z. 71 (1959), 380398. MR 0106918 (21:5648)
 [3]
 , On the free product of associative rings. II, Math. Z. 73 (1960), 433456. MR 0113916 (22:4747)
 [4]
 , On the free product of associative rings. III, J. Algebra 8 (1968), 376383. MR 0222118 (36:5170)
 [5]
 A. Lichtman and W. S. Martindale, The extended center of the coproduct of rings over a division ring, Comm. Algebra 12 (1984), 20672080. MR 747218 (85j:16021)
 [6]
 , The normal closure of the coproduct of domains over a division ring, Comm. Algebra 13 (1985), 16431664. MR 790402 (87b:16003)
 [7]
 W. S. Martindale, The extended center of coproducts, Canad. Math. Bull. 25 (1982), 245248. MR 663623 (83h:16005)
 [8]
 W. S. Martindale and S. Montgomery, The normal closure of coproducts of domains, J. Algebra 82 (1983), 117. MR 701033 (84k:16003)
 [9]
 S. Montgomery, Automorphism groups of rings with no nilpotent elements, J. Algebra 60 (1979), 238248. MR 549109 (81j:16040)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
16A06,
16A03,
16A08
Retrieve articles in all journals
with MSC:
16A06,
16A03,
16A08
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608149241
PII:
S 00029947(1986)08149241
Article copyright:
© Copyright 1986
American Mathematical Society
