ABSTRACT
Much of classical multiplicative ideal theory for integral domains can be extended to commutative rings with zero divisors. This chapter shows how certain results on factorization in integral domains can be extended to commutative rings with zero divisors. It studies factorization in modules and considers various definitions of associates and primitive elements in modules. The chapter discusses the factorization properties of the regular elements (i.e., nonzero divisors) of a commutative ring and exploits the highly developed theory of factorization in commutative cancellative monoids. There are basically two approaches to generalizing results about integral domains to commutative rings with zero divisors. The first is to directly extend a definition concerning elements or ideals of an integral domain to all the elements or ideals of a commutative ring. The second approach is to extend a definition concerning nonzero elements or ideals of an integral domain to just the regular elements or regular ideals of a commutative ring.
