Let L be a left ideal of a right alternative ring A with characteristic ::/=2. If L is maximal and nil, then L is a two-sided ideal. If L is minimal, then it is either a two-sided ideal, or the ideal it generates is contained in the right nucleus of A. In particular, if A is prime, then a minimal left ideal of A must be a two-sided ideal. Let A be a semiprime locally (-1, 1) ring with characteristic ::1=2, 3. Then A is isomorphic to a subdirect sum of an alternative ring, a strong (-1, 1) ring, and a locally (-1, 1) ring with a certain property, where each of these three rings is likewise semiprime with characteristic ::/=2, 3, and where each satisfies any minimal condition satisfied by A. In particular, when A satisfies the minimal condition on two-sided ideals, then A satisfies certain additional identities; and when A satisfies the minimal condition on right ideals, then it must be alternative.