Various results about (von Neumann) regular rings and their projective modules are carried over to ideals in regular rings. These include relations among the concepts of one-sided unit-regularity, stable rank \(1\), separativity, cancellation and substitution properties. In particular, the authors define a condition they call `the comparability' for an ideal \(I\) in a regular ring \(R\) (it is actually a relative one-sided unit-regularity condition), and prove that it is equivalent to (a) certain relative stable rank \(1\) conditions; (b) one-sided unit-regularity for all corners \(eRe\subseteq I\); (c) substitution conditions for \(R\)-module decompositions \(M=R_1\oplus B_1=R_2\oplus B_2\) such that \(R_1\cong R_2\cong R\) and the composition \(R @>\cong>>R_1@>\text{inj}>>M@>\text{proj}>>R_2@>\cong>>R\) is given by an element of \(1+I\). They also show that a minimal ideal \(I\) satisfies `the comparability' if and only if it is separative. Finally, they extend \textit{R. E. Hartwig}'s version of Roth's equivalence theorem [Proc. Am. Math. Soc. 59, 39-44 (1976; Zbl 0347.15005)] from unit-regular rings to unit-regular ideals of regular rings. The reader should be warned that the authors' notion of `the comparability' has little to do with any comparability condition on modules. In particular, when \(I=R\), `the comparability' for \(R\) is just the condition of one-sided unit-regularity. While any regular ring satisfying the (standard) comparability axiom (i.e., for any finitely generated projective modules \(A\) and \(B\), either \(A\) embeds in \(B\) or vice versa) must be one-sided unit-regular, most one-sided unit-regular rings -- and even most unit-regular rings -- fail to satisfy the comparability axiom.