Let R be a commutative ring with 1. By keeping track of left and right modules, this
can also be done for noncommutative rings. Let M, N, B be R-modules.
Definition. A function f : M N B is bilinear if for all m, mi M, n, ni N, r R,
Modules over a PID: Applications to matrices
We assume all the usual facts about polynomial rings over a field as well as basic linear
algebra. From this we develop the Jordan and rational canonical forms for matrices. The
basic idea is to find special fo
Topics in commutative ring theory
From now on all rings are commutative with 1.
Definition. The nilradical of a ring R, denoted Nil R, is the set of all nilpotent elements
Notice that Nil R is an ideal since xm = 0 = y n = (x + y)n+m = 0.
Bilinear and Quadratic Forms
In this chapter, F will always denote a field of characteristic different from 2 and F will
denote the multiplicative group of nonzero elements.
Definition. An n-ary (or rank n) quadratic form q over F is a polynomial in n var
Semisimple Rings and Modules
For this section, R will denote any ring with 1.
Definition. The Jacobson radical of R, denoted Rad R, is the intersection of all maximal
left ideals of R.
Examples: Rad Z = cfw_0 and Rad Z/(8) = (2).
Proposition 1. x Rad R if
Lattices and Boolean algebras
This includes topics of combinatorial interest, with applications in many areas including
computer science and logic. The areas are well represented in the UH Math Department.
Definition. A poset or partially ordered set is a