Contemporary Abstract Algebra, Eighth Edition, by Joseph A. Gallian
Many sets are naturally endowed with two binary operations: addition and multiplication. Examples that quickly come to mind 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. This notion was originated in the mid-19th century by Richard Dedekind, although its first formal abstract definition was not given until Abraham Fraenkel presented it in 1914.
Definition Ring
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:
- $a +b = b + a$
- $(a + b) + c = a + (b + c)$
- There is an additive identity $0$. That is, there is an element 0 in R such that $a + 0 = a$ for all a in R.
- There is an element $-a$ in R such that $a + (-a)=0$.
- $a(bc)=(ab)c$
- $a(b+c)=ab+ac$ and $(b +c) a=ba+ca$.
So, a ring is an Abelian group under addition, also having an associative multiplication that is left and right distributive over addition. Note that multiplication need not be commutative.
When it is, we say that the ring is commutative. Also, a ring need not have an identity under multiplication. A unity (or identity) in a ring is a nonzero element that is an identity under multiplication. A nonzero element of a commutative ring with unity need not have a multiplicative inverse. When it does, we say that it is a unit of the ring. Thus, a is a unit if $a^{-1}$ exists.
The following terminology and notation are convenient. If a and b belong to a commutative ring R and a is nonzero, we say that a divides b (or that a is a factor of b) and write $a | b,$ if there exists an element $c$ in R such that $b=ac$. If a does not divide b, we write a b.
Recall that if a is an element from a group under the operation of addition and n is a positive integer, $na$ means $a +a +…+ a$, where there are n summands. When dealing with rings, this notation can cause confusion, since we also use juxtaposition for the ring multiplication. When there is the potential for confusion, we will use $n . a$ to mean $a+a +…+a$ (n summands).
