After a brief preamble on hereditary rings (Section 2.1), this chapter introduces our main topic, free ideal rings (firs) which form a generalization of principal ideal domains (Section 2.2 and 2.3); frequently they satisfy a weak algorithm relative to a filtration (Section 2.4), which generalizes the division algorithm (relative to a degree-function), to which it reduces in the commutative case. The most important example is the free associative algebra over a field K, characterized as a filtered K-algebra with weak algorithm in Section 2.5, while a useful invariant, the Hilbert series, is described in Section 2.6. Some consequences of the weak algorithm are traced out in Section 2.7 and 2.8; the inverse weak algorithm, using a generalization of the order-function, is used to describe power series rings in Section 2.9 and a transfinite form of the weak algorithm is applied in Section 2.10 to construct one-sided examples. In Section 2.11 a method is described which in many cases allows one to read off from the presentation of a ring whether the n-term weak algorithm holds. This enables one to construct quite naturally n-firs that are not (n + 1)-firs.