Idealization of a Module

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 results about 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 .