Nearrings of continuous functions from topological spaces into topological nearrings

Let lambda be a map from the additive Euclidean n-group R(n) into the space R of real numbers and define a multiplication * on R(n) by v * w = (lambda(w))v. Then (R(n), +, *) is a topological nearring if and only if lambda is continuous and lambda(av) = a lambda(v) for every v is an element of R(n) and every a in the range of lambda. For any such map lambda and any topological space lambda we denote by N-lambda(X, R(n)) the nearring of all continuous functions from X into (R(n), +, *) where the operations are pointwise. The ideals of N-lambda(X, R(n)) are investigated in some detail for certain X and the results obtained are used to prove that two compact Hausdorff spaces X and Y are homeomorphic if and only if the nearrings N-lambda(X, R(n)) and N-lambda(Y, R(n)) are isomorphic.