Rings

6. Rings of sets: The idea of forming a ring from operations on sets is due to George Boole, who published in 1854 An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities. Boole approached logic in a new way reducing it to algebra, in much the same way as Descartes had reduced geometry to algebra.
The familiar set operations of union and intersection satisfy some but not all of the ring axioms. They are both commutative and associative, and satisfy the distributive laws both ways round; but they do not satisfy the identity and inverse laws for addition.
Boole’s algebra of sets works as follows. Let P(A), the power set of A, be the set of all subsets of the set A. Now we define addition and multiplication on P(A) to be the operations of symmetric difference and intersection respectively:
x y = x4y, xy = x\y.

A ring satisfying the further condition that xx = x for all x is called a Boolean ring.

7. Zero rings: Suppose that we have any set R with a binary operation satisfying the additive axioms (A0)–(A4). (We will see later in the course that such a structure is called an abelian group.) Then we can make R into a ring by defining xy = 0 for all x,y ∈   R. This is not a very exciting rule for multiplication, but it is easy to check that all remaining axioms are satisfied.
A ring in which all products are zero is called a zero ring. It is commutative, but doesn’t have an identity (if |R| > 1).

8. Direct sum: Let R and S be any two rings. Then we define the direct sum RS as follows. As a set, R⊕ S is just the cartesian product R⊕S. The operations are given by the rules
(r1, s1) (r2, s2) = (r1 r2, s1 s2), (r1, s1)(r2, s2) = (r1r2, s1s2).
(Note that in the ordered pair (r1 r2, s1 s2), the first denotes addition in R, and the second is addition in S.)

9. Modular arithmetic: Let Zn denote the set of all congruence classes modulo n, where n is a positive integer. We saw in the first chapter that there are n congruence classes; so Zn is a set with n elements:
Zn = {[0]n, [1]n, . . . , [n−1]n}.
Define addition and multiplication on Zn by the rules
[a]n [b]n = [a b]n, [a]n[b]n = [ab]n.
There is an important job to do here: we have to show that these definitions don’t depend on our choice of representatives of the equivalence classes.

10. Rings of functions: The sum and product of continuous real functions are continuous. So there is a ring C(R) of coninuous functions from R to R, with
( f g)(x) = f (x) g(x), ( f g)(x) = f (x)g(x).
There are several related rings, such asC1(R) (the ring of differentiable functions), C0(R) (the ring of continuous functions satisfying f (x) ! 0 as x ! ±¥), and
C([a,b]) (the ring of continuous functions on the interval [a,b]. All these rings are commutative, and all except C0(R) have an identity (the constant function with value 1).
These rings are the subject-matter of Functional Analysis.