Maria Agnesi

Let a and b are two nonzero elements of a ring R such that ab=0. Then a and b are said to be divisors of zero (or zero divisors). In the ring Z_n, the divisors of zero are precisely those nonzero elements that are not relatively prime to n. As a corollary we have that if p is a prime then Z_p has no divisors of zero. Note further that the cancellation laws hold in a ring R if and only if R has no divisors of zero.

A ring in which the multipliation is commutative is called a commutative ring. A ring with a multiplicative identity element is a ring with unity; the multiplicative identity element 1 is called an unity. A multiplicative inverse of an element a in a ring R with nonzero unity 1 is an element a'∈R such that aa' = 1.

Let R be a ring with nonzero unity 1. An element u in R is a unit of R if it has a multiplicative inverse in R. If every nonzero element of R is a unit, then R is called division ring (or skew field). A field is a commutative division ring. A commutative division ring is called a strictly skew field.

By an integral domain, we mean that a commutative ring with a nonzero unity and containing no divisors of zero. We easily see that every field is an integral domain. Furthermore, we see that:

Theorem 1  Every finite integral domain is a field.

Proof. Let 0, 1, a₁, a₂, …, a_n be all the elements of a finite domain D. We need to show that for every a∈D where a≠0, there exists b∈D such that ab=1. Now consider  a1, aa₁, aa₂, …, aa_n. We claim that all these elements of D are distinct, for aa_i=aa_j implies that a_i=a_j by the cancellation loaws that hold inan integral domain. Also, since D has no zero divisors, none of these elements is zero. Hence by counting, we find that a1, aa₁, aa₂, …, aa_n are elements 1, a₁, a₂, …, a_n in some order, so that either a1=1, that is a-1, or aa_i=1 for some i. Thus a has a multiplicative inverse. QED.

Corollary 2  If p is a prime, then Z_p is a field.

Remark: Corollary 2 can be derived from Fermat's Little Theorem. (See 70th article.)

In fact the antecedent of Theorem 1 can be weaken as follows:

Theorem 3 (Wedderburn)  Every finite division ring is a field.

The proof of Theorem 3 is too long to write in HTML, so I attach a PDF file of the proof:

Agnesi_090304_a_finite_division_ring_is_a_field.pdf

References:
Homepage of DEPT of Mathematics in Colgate University
John B. Fraleigh, A First Course in Abstract Algebra 7ed, Addison-Wesley

Posted by Maria Agnesi