: Let R be a ring. A left ideal J in R is called regular if there exists u ? R such that xu - x ? J for each x ? R. The element u is also called a modular right unit for J, and J is also called modular left ideal. Let X be a topological space and let (X) be the space of all continuous bonded functions on X. We will say that (A, ) or simply (A, ? . ?) is a Banach algebra in (X) if A is a subalgebra of (X) and A is a Banach algebra under pointwise multiplication and its own norm ?? . ?. The algebra A is said to be regular on X if for each closed subsed E of X and each x ? X\\E, there is a function f ? A such that f(E) ? {0} and f(x) = 1. For each x ? X consider the following sets = {? ? A : ?(x) = 0} and = {m ? : (m, ) = 0}. Then eing the kernel of the second adjoint of the evaluation character f ?f(x) on A is a maximal -closed ideal i .