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Type Radicals

Published online by Cambridge University Press:  18 May 2009

W. G. Leavitt
W. G. Leavitt
Affiliation:
University of Nebraska
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The lower radical of a module type. For a ring R with unit, the module type t(R) was defined in [6] as follows: t(0) = 0; t(R) = d if every free R-module has invariant rank; t(R) = (c, k) for integers c, k ≧ 1 if every free R-module of rank < c has invariant rank, while a free module of rank h ≧ c has rank h + nk for any integer n ≧ 0. The module types form a lattice under the ordering 0 < (c, k) < d and (c', k') ≦ (c, k) if and only if c ' ≦ c and k' Ⅰ k. Two of the basic theorems on types are:

A. [6; Theorem 2, p. 115] If R → R' is a unit-preserving homomorphism, then t(R')≦ t(R).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 9 , Issue 1 , January 1968 , pp. 22 - 29
Copyright
Copyright © Glasgow Mathematical Journal Trust 1968

References

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