Polyclones over valuation domains.

A clone over a valuation domain is a divisible torsion uniserial module whose non-zero elements have principal annihilator. A clone is standard if it is isomorphic to , where is the field of fractions of , otherwise it is non-standard.

A polyclone is an -module which is the union of a pure composition series whose composition factors are clones. A crucial result for the study of polyclones is that is not zero for all clones . We present a recent proof of this fact obtained just using homological algebra and the structure of the semiring of polyclones equipped with the direct sum and the product induced by the functor .