During this publication the authors enhance a thought of unfastened noncommutative features, in either algebraic and analytic settings. Such capabilities are outlined as mappings from sq. matrices of all sizes over a module (in specific, a vector house) to sq. matrices over one other module, which admire the scale, direct sums, and similarities of matrices. Examples comprise, yet are usually not restricted to, noncommutative polynomials, energy sequence, and rational expressions. Motivation and idea for utilizing the idea of loose noncommutative services frequently comes from loose likelihood. an incredible software quarter is "dimensionless" matrix inequalities; those come up, e.g., in a number of optimization difficulties of procedure engineering. between different comparable parts are these of polynomial identities in earrings, formal languages and finite automata, quasideterminants, noncommutative symmetric capabilities, operator areas and operator algebras, and quantum keep watch over.