Ideals and quotient rings

An ideal I in a commutative ring R is an additive subgroup that absorbs multiplication by arbitrary elements of R. In particular, it contains zero, is closed under differences, and contains ri whenever i belongs to I and r belongs to R. Multiples of a fixed element form a principal ideal, written with parentheses around that generator. An ideal generated by several elements contains all finite ring-linear combinations of those generators.

The even integers form an ideal in the integers, while the integers viewed inside the rationals do not form an ideal there: multiplying the integer one by the rational one-half leaves the subset. Absorption depends on the ambient ring. Also, an ideal containing a unit must be the whole ring, because multiplying that unit by its inverse puts one in the ideal, and multiplying one by any ring element then puts everything in it.

Declare a and b equivalent when their difference belongs to I. The resulting class a plus I contains every element that differs from a by something ignored. Add classes by adding representatives, and multiply classes by multiplying representatives. These recipes are useful only if a different choice from each class gives the same output class. Additive subgroup closure settles addition, while the extra absorption condition settles multiplication.

To check multiplication, replace a by a plus i and b by b plus j, with i and j in I. The new product differs from ab by aj plus bi plus ij. Every term belongs to I by absorption, so the output class is unchanged. This calculation is the purpose of the ideal axioms. Quotienting by an arbitrary additive subgroup can fail: integers inside rationals form an additive subgroup, but multiplying congruent representatives by one-half does not respect the proposed equivalence.

In the quotient of real polynomials by the ideal generated by x squared plus one, the class of x squares to negative one. Polynomial division gives every class a unique representative of degree at most one, so its elements look like a plus bx. With the imposed relation, multiplication reproduces complex arithmetic. The relation holds between classes; it does not claim that the formal polynomial x squared plus one was the zero polynomial before quotienting.

Change the ideal to the one generated by x squared and the resulting quotient behaves differently. The class of x is nonzero, since x is not a multiple of x squared, yet its square is zero. Such an element is nilpotent and is a zero divisor. A quotient of a domain need not remain a domain. Studying the defining ideal, rather than relying on the original ring's familiar behavior, tells us which properties survive.