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With the goal of computing the Grothendieck group of certain multigradedinfinite polynomial rings and the $K$-series of infinite matrix Schubertspaces, we introduce a new type of $\Gamma$-graded $k$-algebra (which we call aPDCF algebra) and a new type of graded module (a BDF module) over said algebra.Since infinite polynomial rings are not Noetherian and the modules of interestare not finitely generated, we compensate by requiring certain finitenessproperties of the grading group $\Gamma$ and of the grading itself. If $R$ is aPDCF $\Gamma$-graded $k$-algebra, we prove that every projective BDF$\Gamma$-graded $R$-module is free, and that when the graded maximal ideal of$R$ is generated by a regular sequence, a BDF analog of the Hilbert SyzygyTheorem holds: every BDF $R$-module has a free resolution of BDF $R$-moduleswhich, though not finite, has the property that each graded piece is eventuallyzero. We use this to show that the Grothendieck group of projective BDF$R$-modules (defined similarly to $K_0(R)$) is isomorphic to the Grothendieckgroup of all BDF $R$-modules (defined similarly to $G_0(R)$). We describe thisGrothendieck group explicitly by using $K$-series to give an isomorphism with acertain space of formal Laurent series. Finally we give a BDF version ofSerre's formula for the product in the Grothendieck group, making it into aring.

The Grothendieck Ring of Certain Non-Noetherian Group-Graded Algebras via $K$-Series