ABSTRACT
Here we introduce some basic terminology, and give a sample of a modern construction of a universal object, namely a polynomial ring in one variable.
The idea of ring generalizes the idea of collection of numbers, among other things, so maybe it is a little more intuitive than the idea of group. A ring R is a set with two operations, + and ·, and with a special element 0 (additive identity) with most of the usual properties we expect or demand of addition and multiplication: • R with its addition and with 0 is an abelian group. [1] • The multiplication is associative: a(bc) = (ab)c for all a, b, c ∈ R. • The multiplication and addition have left and right distributive properties: a(b + c) = ab+ ac and (b+ c)a = ba+ ca for all a, b, c ∈ R. Often the multiplication is written just as juxtaposition
ab = a · b
Very often, a particular ring has some additional special features or properties:
• If there is an element 1 in a ring with 1 · a = a · 1 for all a ∈ R, then 1 is said to be the (multiplicative) identity or the unit [2] in the ring, and the ring is said to have an identity or have a unit or be a ring with unit. [3]
• If ab = ba for all a, b in a ring R, then the ring is a commutative ring. That is, a ring is called commutative if and only if the multiplication is commutative.
