Inverness-shire Inverness-shire is Scotland's largest county, and the second largest in the UK as a whole after Yorkshire. It borders Ross to the north, Nairnshire, Moray, Banffshire and Aberdeenshire to the east, and Perthshire and Argyll to the south. It covers a large mainland area and various island areas off the west coast. The mainland area has coastline in both the east and the west. The island areas include North Uist, South Uist and Harris in the Outer Hebrides and Skye and the Small Isles in the Inner Hebrides. Principal settlements include Inverness, Fort William, Fort Augustus and Portree. Inverness-shire's coat
Principal ideal ring Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains. A commutative principal ideal ring which is also an integral domain is said to be a "principal ideal domain" (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain. If "R" is a right principal ideal ring, then it is certainly a right Noetherian ring, since every right ideal is finitely generated. It is also a right Bézout ring since all finitely generated right ideals are principal. Indeed, it is clear that