What is a groupin abstract algebra?
A group, usually referred to as G,is a finite or infinite set of elements together with a
binary operation (often called the group operation) that together satisfy the four fun-
damental properties—closure, associativity, and the identity and inverse properties
(for more information about these properties, see elsewhere in this chapter). A great
many of the objects investigated in mathematics turn out to be groups, including
familiar number systems—such as the integers, rational, real, and complex numbers
under addition; non-zero rational, real, and complex numbers under multiplication;
non-singular matrices under multiplication; and so on. The branch of mathematics
that studies groups is called group theory, an important area of mathematics that has
many applications to mathematical physics (such as particle theory).
What is a ring?
A ring is an algebraic structure (some definitions say a set) in which two binary opera-
tors (addition and multiplication) in various combinations must satisfy either the
additive associative, commutative, identity, and inverse properties, the multiplicative
associative property, or the left and right distributivity properties. For example, the
elements of one operation, such as addition, must form a group that is commutative,
also known as an abelian group. The multiplicative operation must produce unique
answers that have the associative property. These two operations are further connect-
ed by requiring the multiplication to have a distributive property with respect to the
addition. This can be written as follows, with a, b,and celements of the ring:
a(bc) (ab)(ac) and (bc) a(ba) (ca) 161ALGEBRA
Does everyone agree with axioms?
N
o, not everyone agrees with all axioms, the self-evident truth upon which
knowledge must rest and other knowledge is built. For example, not all episte-
mologists (philosophers who deal with the nature, origin, and scope of knowledge)
agree that any true axioms exist. However, in mathematics axiomatic reasoning is
widely accepted, where it means an assumption on which proofs are based.The word axiom (or postulate) comes from the Greek word axiomaand
means “that which is deemed worthy or fit,” or “considered self-evident.” Ancient
Greek philosophers used the term axiom as a claim that was true without any
need for proof. In modern mathematics, an axiom is not a proposition that is self-
evident but simply means a starting point in a logical system. For example, in
some rings (see below), the operation of multiplication is commutative (said to
satisfy the “axiom of communtativity of multiplication”), and in some it is not.