Modules, Homomorphisms

Homomorphisms

Review the general definition of a homomorphism. As you'd expect, a module homomorphism is a function f that maps one module into another, such that f commutes with group addition and scaling by R. In other words, f is a group homomorphism, and f(x*a) = x*f(a).

For instance, let M = R[x], the polynomials in x with coefficients from R. Polynomials are added as usual, and scaled by elements of R on the left. Let f be a module homomorphism that multiplies polynomials by x. It commutes with addition and scaling. Note that f is not a ring homomorphism. That is, f(x)×f(x) is not f(x²).

The kernel of a module homomorphism is a submodule. One can take the quotient over such a submodule, adding cosets as expected, and letting a ring element act on a coset by allowing it to act on any coset representative. Like groups and rings, the image of a module homomorphism is isomorphic to the cosets of the kernel. The correspondence theorems hold as well, so that submodules in the domain containing the kernel correspond 1-1 with submodules in the range.

The only new twist is that every submodule is normal. This is because modules are based on abeliang groups, and subgroups of abelian groups are always normal.

Also, homomorphisms from one module to another can be added by (f+g)(a) = f(a)+g(a). This gives another module homomorphism. Thus module homomorphisms form an abelian group under addition. The zero homomorphism maps the entire module onto 0, the group identity.

If a module M is split into U and V, where U is unitary, any homomorphism on M can be restricted to U and V, and the resulting images lie in the corresponding submodules in the range. If x is in U, 1*f(x) = f(1*x) = f(x), hence f(x) is in the unitary component of the image. Similarly, for x in V, 1*f(x) = f(1*x) = f(0) = 0. If f is onto, or 1-1, then the restriction of f to U and V is onto, or 1-1.

In fact the function f defines, and is defined by, its action on U and V. The homomorphism is the cross product of the component homomorphisms on U and V. The group of homomorphisms is the direct product of the groups of homomorphisms on U and V. If you're not following all this, that's probably ok; we spend most of our time working with unitary modules anyways. Unless otherwise specified, the term "module" means a unitary module, just as a "ring" means a ring with 1.