1. FAILURE OF CANCELLATION FOR QUARTIC AND HIGHER-DEGREE ORDERS
- Author
-
Ryan Karr
- Subjects
Combinatorics ,Discrete mathematics ,Algebra and Number Theory ,Field extension ,Residue field ,Applied Mathematics ,Order (group theory) ,Principal ideal domain ,Field (mathematics) ,Quotient ,Prime (order theory) ,Mathematics ,Separable space - Abstract
Let D be a principal ideal domain with quotient field F and suppose every residue field of D is finite. Let K be a finite separable field extension of F of degree at least 4 and let [Formula: see text] denote the integral closure of D in K. Let [Formula: see text] where f ∈ D is a nonzero nonunit. In this paper we show, assuming a mild condition on f, that cancellation of finitely generated modules fails for R, that is, there exist finitely generated R-modules L, M, and N such that L ⊕ M ≅ L ⊕ N and yet M ≇ N. In case the unit group of D is finite, we show that cancellation fails for almost all rings of the form [Formula: see text], where p ∈ D is prime.
- Published
- 2002