{0} ⊂ T is a sub-module. It is called the to sion submodule of M. Proof. Because a domain has no zero-divisors, we can conclude that: a1t1 = 0 and a2t2 = 0 implies a1a2(t1 + t2) = 0 and a1, a2 6= 0 …

The resulting R-module M=N is called the quotient module of M with re-spect to the submodule N. The noether isomorphism theorems, which we have seen previously for groups and rings, then have …

We’ll later see how to understand module by looking at generators and relations — this turns out to be easier than the corresponding problem for a group. But first we’ll look at another example of a …

All in all the approach chosen here leads to a clear refinement of the customary module theory and, for M = R, we obtain well-known results for the entire module category over a ring with unit.

Modules are best initially thought of as abelian groups with additional structure. In particular, we would expect most of the basic facts we derived earlier for groups (hence for abelian groups) to hold true.

If S is a subring of R then any R-module can be considered as an S-module by restricting scalar multiplication to S M. For example, a complex vector space can be considered as a real vector space …

As before, the basic examples are OX (a left D-module), X (a right D-module), DX (both a left and a right D-module). We see that the notion of a D-module on X is local.