Idealization of a module

Abstract

Let R be a commutative ring and M an Rmodule. Nagata introduced the idealization R(+)M of M. Here R(+)M = R ⊕ M (direct sum) is a commutative ring with product (r1, m1)(r2, m2) = (r1r2, r1m2 + r2m1). The name comes from the fact that if N is a submodule of M, then 0 ⊕ N is an ideal of R(+)M. The idealization can be used to extend resultsabout ideals to modules and to provide interesting examples of commutative rings with zero divisors. We survey known results concerning R(+)M and give some new ones too. The theme throughout is how properties of R(+)M are related to those of R and M. © 2009 Rocky Mountain Mathematics Consortium. All rights reserved.