Abstract Algebra: Theory and Applications · Problem SE.20.4
Fields — Thomas Judson
Mathematics — Abstract Algebra · modules
Problem
What happens if we try to do linear algebra over a ring that is not also a field? The object that resembles a vector space, but with this one distinction, is known as a module. You can build one easily with a construction like \texttt{ZZ\^3}. Evaluate the following to create a module and a submodule.
\begin{verbatim}
M = ZZ^3
u = M([1, 0, 0])
v = M([2, 2, 0])
w = M([0, 0, 4])
N = M.submodule([u, v, w])
\end{verbatim}
Examine the bases and dimensions (aka ``rank'') of the module and submodule, and check the equality of the module and submodule. How is this different than the situation for vector spaces? Can you create a third module, P, that is a proper subset of M and properly contains N?