An element u in an associative ring R with identity is normalising if \(Ru=uR\). A quasi-commutative principal ideal ring (QCPIR) is a ring in which every two-sided ideal is generated by a normalising element. The author develops a detailed structure theory for QCPIRs, including the following results: any QCPIR is uniquely a finite direct sum of indecomposable QCPIRs, and each indecomposable factor is prime or satisfies d.c.c. on two-sided ideals, or both. A QCPIR may be embedded in a ring of fractions (which is a QCPIR with d.c.c. on two-sided ideals). If A is a nonzero ideal of an indecomposable QCPIR, other than the unit ideal, then A is a unique product of nonzero prime ideals. Any non- minimal prime ideal of a QCPIR is maximal. A QCPIR may be left- or right- primitive but not simple, and there exist indecomposable QCPIRs with d.c.c. on two-sided ideals which are not homomorphic images of prime QCPIRs.