Coproducts and some universal ring constructions
Author:
George M. Bergman
Journal:
Trans. Amer. Math. Soc. 200 (1974), 3388
MSC:
Primary 16A64
MathSciNet review:
0357503
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Abstract: Let be an algebra over a field , and be two nonzero finitely generated projective modules. By adjoining further generators and relations to , one can obtain an extension of having a universal isomorphism of modules, . We here study this and several similar constuctions, including (given a single finitely generated projective module ) the extension of having a universal idempotent moduleendomorphism , and (given a positive integer ) the algebra with a universal algebra homomorphism of into its matrix ring, . A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring ( , and respectively in the above cases), and hence to apply the theory of such coproducts. As in that theory, we find that the homological properties of the construction are extremely good: The global dimension of is the same as that of unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relation . These results allow one to construct a large number of unusual examples. We discuss the problem of obtaining similar results for some related constructions: the adjunction to of a universal inverse to a given homomorphism of finitely generated projective modules, , and the formation of the factorring by the trace ideal of a given finitely generated projective module (in other words, setting ). The idea for a categorytheoretic generalization of the ideas of the paper is also sketched.
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 [1]
 S. A. Amitsur, Embeddings in matrix rings, Pacific J. Math. 36 (1971), 2129. MR 43 #2017. MR 0276270 (43:2017)
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 Hyman Bass, Algebraic theory, Benjamin, New York, 1968. MR 40 #2736. MR 0249491 (40:2736)
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 George M. Bergman, Hereditary commutative rings and centres of hereditary rings, Proc. London Math. Soc. (3) 23 (1971), 214236. MR 46 #9022. MR 0309918 (46:9022)
 [4]
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 , Rational relations and rational identities in division rings I, J. Algebra (to appear). (Note: [9] is a prerequisite to [5].) MR 0432697 (55:5683a)
 [6]
 , Some examples in p.i. ring theory, Israel J. Math. (to appear).
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 , Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 132. MR 0357502 (50:9970)
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 George M. Bergman and Lance W. Small, P.i degrees and prime ideals (to appear). MR 0360682 (50:13129)
 [10]
 A. I. Bowtell, On a question of Mal'cev, J. Algebra 7 (1967), 126139. MR 37 #6310. MR 0230750 (37:6310)
 [11]
 W. Edwin Clark and George M. Bergman, The automorphism class group of the category of rings, J. Algebra 24 (1973), 8099. MR 0311648 (47:210)
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 P. M. Cohn, Universal algebra, Harper and Row, New York, 1965. MR 31 #224. MR 0175948 (31:224)
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 , Some remarks on the invariant basis property, Topology 5 (1966), 215228. MR 33 #5676. MR 0197511 (33:5676)
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 , Localization in semifirs (to appear).
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 A. A. Klein, A remark concerning embeddability of rings in fields, J. Algebra 21 (1972), 271274. MR 45 #8670. MR 0299622 (45:8670)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403575037
PII:
S 00029947(1974)03575037
Keywords:
Universal module homomorphism (isomorphism),
finitely generated projective module,
matrix ring,
coproduct of rings,
representable functor,
semigroup of modules (resp. projective modules) under ``",
global dimension,
fir,
localization,
trace ideal of a projective module,
linear category
Article copyright:
© Copyright 1974
American Mathematical Society
