SOME FACTORIZATION PROPERTIES OF IDEALIZATION IN COMMUTATIVE RINGS WITH ZERO DIVISORS

We study some factorization properties of the idealization R(+)M of a module M in a commutative ring R which is not necessarily a domain. We show that R(+)M is ACCP if and only if R is ACCP and M satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which R(+)M is a BFR. We also characterize the idealization rings which are UFRs.