Mathematics Prelims

December 1, 2009

Submodules and Homomorphisms

Filed under: Abstract Algebra,Algebra — cjohnson @ 8:57 pm

Just as a subgroup is a group within a group or a subfield is a field within a field, a submodule is a module within a module.  That is, if M is an R-module, we say that N is a submodule of M if N is a subgroup of M and is closed under multiplication by elements of R.  This can be summarized by saying that for every x, y \in N and every r \in R, the following two properties hold.

  1. x - y \in N
  2. rx \in N

In the case of groups, we could only form a quotient group if the subgroup we were modding out by as sufficiently “nice” (i.e., was a normal subgroup); likewise, in the case of rings we required that a subring be nice (an ideal) in order to form a quotient ring.  With modules however, we can always form the quotient module with a submodule.  We can do this since, as M is an abelian group under addition, all subgroups are normal and we can form the quotient group M/N.  This is naturally an abelian group, so in order to turn this into an R-module we have to define multiplication of elements in R and M/N, which we do in the most obvious way: r(x + N) = rx + N.  To check that this is in fact an R-module, we simply verify that the three axioms of a module hold.

  1. r((x + N) + (y + N)) = r(x + y + N) = r(x + y) + N = rx + ry + N = rx + N + ry + N
  2. (r + s)(x + N) = (r + s)x + N = rx + sx + N = rx + N + sx + N
  3. (rs)(x + N) = (rs)x + N = r(sx) + N = r(sx + N)

Finally, if M is a unitary module, then so too is M/N: 1(x + N) = 1x + N = x + N.

We say a map \phi : M \to N between two R-modules is a homomorphism if \phi is a group homomorphism with the additional property that for each r \in R, \phi(rx) = r \phi(x).  As you would expect, the kernel of this homomorphism is a submodule of M, and the image is a submodule of N.

Supposing that S \subseteq M, we define R\langle S \rangle to be the smallest submodule of M containing S, which is naturally the intersection of all submodules containing S

\displaystyle R\langle S \rangle = \bigcap \left\{ N : N \text{ a submodule of } M \text{ and } S \subseteq N \right\}

If S is a finite set, we may write R\langle s_1, s_2, \ldots, s_n \rangle in place of R \langle S \rangle, and in the event that S is a singleton, we say that R\langle s_1 \rangle is a cyclic submodule of M.  We call the elements of S the generating set of the R \langle S \rangle submodule, and call the elements of S the generators of this submodule.

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