On maximal ideals which are also minimal prime ideals in certain Banach rings

We study the existence of a maximal ideal which is also a minimal prime ideal in Banach rings in a wide class containing the Banach algebra C-bd (X, k) of bounded continuous functions X -> k for a topological space X and a Banach field k with a mild condition, the quotient of C-bd (X, k) by the closed ideal C-0 (X, k) of functions vanishing at infinity, the bounded direct product Pi(lambda is an element of Lambda) kappa(lambda) to of a family n = (nA)AE A of Banach fields with a mild condition, and the quotient of Pi(lambda is an element of Lambda) kappa(lambda) by the completed direct sum (circle plus) over cap (lambda is an element of Lambda) kappa(lambda). We describe the maximal spectrum and the Berkovich spectrum of such Banach rings, and generalise the classical result on the relation between the existence of such a maximal ideal of the Banach R-algebra C-bd(N, R)/C-0(N, R) and the existence of a P-point in beta N\N.