Trivial multiple zeta values in Tate algebras

We study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras in positive characteristic introduced by the second author. If the number of variables involved is 'not large' in a way that is made precise in the paper, we can endow the set of trivial multiple zeta values with a structure of module over a non-commutative polynomial ring with coefficients in the rational fraction field over a finite field. We determine the structure of this module in terms of generators and we show how in many cases, this is sufficient for the detection of some linear relations between Thakur's multiple zeta values.