IDEALIZATION OF A MODULE

Let R be a commutative ring and M an R-module. Nagata introduced the idealization R (+) M of M. Here R(+) M - R circle plus M (direct sum) is a commutative ring with product (r(1), m(1))(r(2), m(2)) = (r(1)r(2), r(1)m(2) + r(2)m(1)). The name comes from the fact that if N is a submodule of M, then 0 circle plus 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.