Quotient module

In algebra, a quotient module is a special way to create a new structure from an existing module and one of its smaller parts, called a submodule. This idea is similar to how we can create a new space from a bigger space and a smaller space inside it, known as a quotient vector space. However, quotient modules are different from similar constructions in other areas of math, like rings or groups, because the smaller part used in those cases has special properties that are not needed here.

To build a quotient module, we start with a module A over a ring R and a submodule B inside A. We create a new space called A/B, where we group together elements of A that differ by an element of B. This grouping follows a specific rule: two elements a and b in A are considered the same, or equivalent, if their difference b - a is in B. The resulting groups are called equivalence classes.

We can perform addition and multiplication on these equivalence classes in a way that keeps the module properties. This means that the quotient module A/B behaves just like the original module A, but with the submodule B essentially "collapsed" to a single point. The map that sends each element a in A to its equivalence class a + B is called the quotient map or projection map, and it helps us understand how the quotient module relates to the original one.

Examples

Imagine you have a set of polynomials, which are expressions like (X^2 + 3X + 2), where the numbers are real. These polynomials form a module. Now, let’s focus on a special subset of these polynomials — those that can be divided by (X^2 + 1) without leaving any remainder. This subset is called a submodule.

When we create a quotient module, we’re grouping these polynomials based on how they behave when divided by (X^2 + 1). Polynomials that give the same remainder after division are considered equivalent. In this quotient module, (X^2 + 1) behaves as if it were zero. The result is a new module that is actually the same as the complex numbers when viewed over the real numbers.