The goal of this chapter is to ensure you aware of the fact that particular rings are not fields (such as integers, polynomial rings, and matrix rings) and that you thotoughly understand what that means – which obviously requires you to know what a ring is in the first place. So, let’s find out.“”‘’
Many sets are naturally endowed with two binary operations: addition and multiplication.
And of course, a binary operation is an operation (or calculation) that applies to two input (or quantities). Examples of such sets are the integers, the integers modulo n, the real numbers, matrices, and polynomials.
When considering these sets as groups, we simply used addition and ignored multiplication. In many instances, however, one wishes to take into account both addition and multiplication. One abstract concept that does this is the concept of a ring. The notion of a ring originated with Richard Dedekind in the mid-nineteenth-century. though the actual term was coined in 1897 by the German mathematician David Hilbert (1862-1943), and its first formal abstract definition was not given until Abraham Fraenkel presented it in 1914.
A ring R is a set with two binary operations, addition (denoted by a + b) and multiplication (denoted by ab), such that for all a, b, c in R:
1. a + b = b + a (commutative under addition)
2. (a + b) + c = a + (b + c) associative (by definition of a group)
3. There is an element 0 in R such that a + 0 = a (by definition of a group)
4. There is an element –a in R such that a + (–a) = 0 (by definition of a group)
5. a(bc) = (ab)c (associative under multiplication)
6. a(b + c) = ab + ac and (b + c)a = ba + ca (the left and right distributive
laws hold for all elements of R