Even if the symbol of the operation is omitted, the order of s and t does matter (unless ∗ is commutative).
A one-sided property is related to one (unspecified) of two sides.
Over non-commutative rings, the left–right distinction is applied to modules, namely to specify the side where a scalar (module element) appears in the scalar multiplication.
The distinction is not purely syntactical because one gets two different associativity rules (the lowest row in the table) which link multiplication in a module with multiplication in a ring.
A bimodule is simultaneously a left and right module, with two different scalar multiplication operations, obeying an associativity condition on them.