WEAKLY (m, n)-CLOSED IDEALS AND (m, n)-VON NEUMANN REGULAR RINGS

Let R be a commutative ring with 1 not equal 0, I a proper ideal of R, and m and n positive integers. In this paper, we define I to be a weakly (m, n)-closed ideal if 0 not equal x(m) is an element of I for x is an element of R implies x(n) is an element of I, and R to be an (m, n)-von Neumann regular ring if for every x is an element of R, there is an r is an element of R such that x(m)r = x(n). A number of results concerning weakly (m, n)-closed ideals and (m, n)-von Neumann regular rings are given.