The multiplicative order convergence was studied and investigated on Riesz algebras. This paper deals with Riesz algebras and different topologies on them. In this paper, we investigate Riesz algebras on which we define various kinds of continuities. We give the relation between them under certain specific conditions. We show some relations among locally full, locally convex and locally solid Riesz algebras. Also, we introduce the notions of order and topological continuity of algebraic multiplications on topological Riesz algebras. Also, we extend the multiplication to quotient spaces of Riesz algebras.