Nonsingular Rings with Finite Type Dimension

Abstract

Two modules are said to be orthogonal if they do not have nonzero isomorphic submodules. An atomic module is any nonzero module whose nonzero submodules are not orthogonal. A module is said to have type dimension n if it contains an essential submodule which is a direct sum of n pairwise orthogonal atomic submodules; If such a number n does not exist, we say the type dimension of this module is ∞. In this paper, we provide characterizations and examples of nonsingular rings with finite type dimension. A characterization theorem is proved for nonsingular rings whose nonzero right ideals contain nonzero atomic right ideals. Type dimension formulas are also obtained for polynomial rings, Laurent polynomial rings and formal triangular matrix rings.KeywordsType DimensionPolynomial RingAtomic ModuleQuotient RingEssential SubmoduleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.