The usual definition of minimal primary decomposition in commutative algebra grants a little too much leeway. Making minimality work for real multiparameter persistence modules -- that is, multigraded modules over rings of polynomials with real exponents -- requires tighter control over what algebraists call socles and computational topologists call deaths. Functorial definitions yield finiteness statements for primary decompositions of reasonable persistence modules. In contrast, for irreducible decompositions finiteness is impossible with continuous parameters; but nonetheless minimality can be characterized by density with respect to certain topologies arising in the context of partially ordered vector spaces.