So far we have focused on groups where only a single binary operation is used. However, there is a second classic algebraic structure that has two binary operations (usually denoted + and *) - namely a ring R. When just considering the first operation, (R, +) is an abelian group. We always denote the identity element under addition by 0. We also require that the ring multiplication * is associative and R satisfies the distributive laws:
for all a, b, c in R. Examples of rings include the integers and real numbers (both under the standard addition and multiplication) and the set of all 2 X 2 matrices with integer entries (under matrix addition and matrix multiplication). Of course, these three examples are all infinite rings. To see a finite example of a ring, consider the integers modulo n. i.e. Zn = {0, 1, 2,..., n-1} for some n >= 2. We already know that (Zn, +) is an abelian group, where + denotes addition mod n. To obtain a ring structure, we need a second binary operation. Let * denote multiplication mod n. For example, if n = 12 then 3 * 8 = 0 and 2 * 9 = 6. Since * is associative and the distributive laws hold for + and *, it follows that (Zn, +, *), which we will denote by Zn, is a ring. Like groups, there are special classes of rings that mathematicians are particularly interested in. For instance, a ring R is commutative if a * b = b * a for all a, b in R. Certainly the integers, real numbers, and Zn all form commutative rings. The ring of 2 X 2 matrices under matrix addition and matrix multiplication does not form a commutative ring. Can you see why? A ring R is said to have unity if there exists some 1R in R such that 1R * a = a * 1R = a. for all a in R. Note that 1R, if it exists, is the multiplicative identity element of R. To introduce the next class of rings we first need a definition. A nonzero element a in R, where R is commutative, is called a zero divisor for R if there exists some nonzero b in R satisfying a * b = 0.
A commutative ring R with unity is called an integral domain if it contains no zero divisors. Note that both the integers and real numbers are integral domains. However, even though Zn is a commutative ring with unity, it is an integral domain only for certain values of n. Question: For which values of n does Zn have zero divisors. (Note: this will determine precisely when Zn is an integral domain). Finally, we define a field to be a commutative ring where the nonzero elements form a group with respect to multiplication. Note that a field necessarily has a unity element 1R. In addition, if a is a nonzero element from a field R, then a must have a multiplicative inverse a-1 in R that satisfies a * a-1 = a-1 * a = 1R. Note that every field is an integral domain. Also, every finite integral domain is a field (both of these results should be in your text as theorems and/or exercises).
The Un applet below allows the user to work with Zn using the operation of multiplication modulo n, i.e. the Zn ring multiplication. Place two elements of Zn in the two seed cells and then graph the image. Do this several times with different seed values and different values of n. What will happen if 0 is placed down one of the sides?
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