Generators of ideals containing monics (Q1085216)

Let R be a commutative unitary ring, and let X be an indeterminate. Let I be an invertible ideal of R[X] which contains a monic. Then there exists a decomposition of R as a direct sum of ideals \(R_ 1,...,R_ k\) so that, for each i, \(IR_ i[X]\) is generated by a monic of \(R_ i[X]\). It follows that \(f\in R[X]\) divides a monic if and only if there exists such a decomposition of R so that, for each i, the component \(f_ i\) of f in \(R_ i[X]\) is of the form \(f_ i=g_ ih_ i\), where \(g_ i\) (respectively, \(h_ i)\) is a monic (respectively, unit) of \(R_ i[X]\). Moreover, f is an associate of a monic if and only if the highest non- nilpotent coefficient of f is a unit of R. The ring R is indecomposable if and only if every invertible ideal of R[X] which contains a monic is generated by a monic. If R is indecomposable, and I, J are comaximal ideals of R[X], then \(I\cap J\) is generated by a \(monic\quad f\) if and only if I, J are generated by monics g, h respectively with \(gh=f.\) Some of the above results (or versions of them) extend to semigroup rings over R.