Carat (mass) quarters" of 15 grains troy each. Likewise, the ounce troy was divisible into 24 "ounce carats" of 20 grains troy each; the ounce carat was divisible into four "ounce grains" of 5 grains troy each; and the ounce grain was divisible into four "ounce quarters" of grains troy each. The "solidus" was also a Roman weight unit. There is literary evidence that the weight of 72 coins of the type called "solidus" was exactly 1 Roman pound, and that the weight of 1 "solidus" was 24 "siliquae". The weight of a Roman pound is generally believed to have been 327.45
Pure subgroup While pure projective modules have not found as many applications as pure injectives, they are more closely related to the original work: A module is pure projective if it is a direct summand of a direct sum of finitely presented modules. In the case of the integers and Abelian groups a pure projective module amounts to a direct sum of cyclic groups. Pure subgroup In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas. A