Multiplicative orders on terms

Let $R$ be a commutative ring with identity. Any order on terms of the polynomial algebra $R[x_1,\dots,x_k]$ induces in a natural way the notion of a leading term. An order on terms is called multiplicative if and only if the leading term of a product equals the product of leading terms. In this paper, we present a procedure for the construction of multiplicative orders. We obtain some characterizations of rings for which such orders exist. We give conditions sufficient for the existence of such orders.