Modules

Many core structures in algebra are richer than groups but poorer than rings. For example, an ideal I ⊂ R in a commutative ring is an additive subgroup, but not a ring because it has no 1 in general. However, one may multiply elements of I with any ring elements. Also, the set M_m,n(R) is an additive group, but not a ring for n ≠ m. Its structure as a cartesian product ring R^mn is rarely considered. But again, one may multiply a matrix by a “scalar” from R. These structures remind us of vector calculus in high school. This is what we now want to investigate for the sake of music theory. The structure of this type is called a “module”, and we want to give a short and very incomplete account of the theory of modules, which plays a major role in mathematical music theory.