Resolving Decompositions for Polynomial Modules

We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases.

Attached prime ideals of generalized inverse polynomial modules

Let [Formula: see text] be a ring, [Formula: see text] a strictly totally ordered monoid which is also artinian and [Formula: see text] a monoid homomorphism. Given a right [Formula: see text]-module [Formula: see text], denote by [Formula: see text] the generalized inverse polynomial module over the skew generalized power series ring [Formula: see text]. It is shown in this paper that if [Formula: see text] is a completely [Formula: see text]-compatible module and [Formula: see text] an attached prime ideal of [Formula: see text], then [Formula: see text] is an attached prime ideal of [Formula: see text], and that if [Formula: see text] is a completely [Formula: see text]-compatible Bass module, then every attached prime ideal of [Formula: see text] can be written as the form of [Formula: see text] where [Formula: see text] is an attached prime ideal of [Formula: see text].