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 principal ideals in formula_43 are of the form formula_44. Actually, every ideal formula_1 in formula_43 is principal, which can be shown in the following way. Suppose formula_47 where formula_48, then consider the surjective homomorphisms formula_49 Since formula_50 is finite, then for sufficiently large formula_51, formula_52 Thus, formula_53, which implies formula_1 is always finitely generated. Since the ideal generated by any integers formula_3 and formula_56, formula_57, is exactly formula_58, by induction on the number of generators, it follows that formula_1 is principal. However, all rings have principal ideals, namely, any ideal generated by exactly one element. For example, the ideal formula_60