[For the entire collection see Zbl 0538.00016.]
Given a multiplicatively closed set T of monic polynomials in A[u], the polynomial ring over a commutative ring A, the author considers the exact category End(A;T) whose objects are pairs \((M,\alpha)\), with M a finitely-generated projective A-module and \(\alpha\) an A-endomorphism of M which satisfies some polynomial in T. There is a forgetful function \((M,\alpha)\mapsto M\) which induces a homomorphism \(K_ i(End(A;T))\to K_ iA\) whose kernel is denoted \(\tilde K_ i(End(A;T)).\) Such groups, for particular choices of T, have turned up in a number of K-theoretic contexts. The author defines various natural operations on these groups and proves a list of relations among these operations. The case of nilpotent endomorphisms \((T=\{u^ m,\quad m\geq 1\})\) receives special attention, as does the relationship with the K-theory of truncated polynomial rings.