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Invertibility and Class Number of Orders

Published online by Cambridge University Press:  20 November 2018

Howard Gorman
Howard Gorman*
Affiliation:
Stanford University, Stanford, California
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This paper continues the work begun in [5] and concerns the invertibility of modules in a finite-dimensional, symmetric algebra L with 1 over a field. In particular, we continue the work done in [5] which dealt with the connection between invertibility in these algebras and a condition called the Brandt Condition, which is a reformulation by Kaplansky [6] of some ideas of Brandt.

We begin by proving some preliminary results on invertibility and some equivalent conditions for the dual of an order to be principal.

Then, we define the class number of an order and reformulate the concept of invertibility in terms of class number. In this terminology, we find some equivalent conditions which ensure that an order in certain algebras L (including commutative, symmetric algebras, and algebras with a strong involution) has class number equal to 1 (i.e., all modules principal), and we characterize a class of Brandt algebras over the quotient fields of valuation rings as those in which all orders have class number less than or equal to 2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 1 - 11
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Dade, E. C., Taussky, O., and Zassenhaus, H., On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field, Math. Ann. 148 (1962), 31–64.Google Scholar
2. Faddeev, D. K., An introduction to the multiplicative theory of modules of integral representations, Trudy Mat. Inst. Steklov. 80 (1965), 145–182.Google Scholar
3. Faddeev, D. K., On the theory of cubic Z-rings, Trudy Mat. Inst. Steklov. 80 (1965), 183–187.Google Scholar
4. Gorman, H. E., Imertibility of modules over Prilfer rings, Illinois J. Math. 14 (1970), 283–298.Google Scholar
5. Gorman, H. E., The Brandt Condition and invertibility of modules, Pacific J. Math. 32 (1970), 351–372.Google Scholar
6. Kaplansky, I., Submodules of quaternion algebras, Proc. London Math. Soc. (3) 19 (1969), 219–232.Google Scholar