Basic Commutative Ring Theory

Here we provide some crucial defininitions and lemmas for the theory of commutative rings. All rings are commutative with unity; all homomorphisms of rings take 1 to 1.

Definition (Integral Domain):

An integral domain is a ring which has no zero divisors. A zero divisor is an element $a$ of a ring $A$ such that there is an element $b$ of $A$ such that $ab=0$.

Definition (Prime Ideal):

A prime ideal $\mf a$ of a ring $A$ is an ideal such that $A/\mf a$ is an integral domain.

Definition (Maximal Ideal):

A maximal ideal $\mf a$ of a ring $A$ is an ideal such that $A/\mf a$ is a field.

The following alterante characterizations are often given as a definition

Lemma: (Alternate Characterization of Prime Ideals):

A proper ideal $\mf a \subsetneq A$ is prime if and only if, for any pair of elements $a, b\in A$, we have that $ab\in A$ implies $a\in A$ or $b\in A$.

And analogously:

Lemma: (Alternate Characterization of Maximal Ideals):

A proper ideal $\mf a \subsetneq A$ is maximal if and only if it is contained in no proper ideal.