Abstract algebra, now commonly referred to as just “algebra,” is a field in mathematics which concerns itself with algebraic structures such as rings, groups, fields, vector spaces, and algebras. These structures bear importance in mathematical physics; for example, quantum mechanics is often concerned with vector spaces (Hilbert space in particular), and general relativity is interested in various Lie algebras. Historically, abstract algebra has been closely related to matrices and linear algebra, functional decomposition and topology.
Common structures in abstract algebra are often related to each other but differ by the addition (or lack) of one or more properties. For example, we can let S represent a set of elements. (Let me be clear in noting that these elements do not necessarily have to be numbers. A set may contain things like shapes or functions.) The set by itself is quite useless and we know nothing of its contents because we do not know in any way how they behave. But if we let set S have an operation, we can learn about the interaction between elements. We can give S a binary operation, or an operation that requires two elements from the set. This operation is commonly referred to as the symbol “+”, but it does not necessarily mean addition in the traditional sense. If the set is closed under this operation (the operation will never result in an element outside the set), the operation is associative, and if we define an identity element such that the operation between any element and the identity is the non-identity element, then we have created a monoid.
As an example, let us take the positive integers from zero to infinity and define the traditional addition as our operation. Standard addition is associative. Adding any two positive integers results in a third positive integer, which is in our set, so we have satisfied closure. We have included zero, our additive identity, which means that any non-zero element plus zero is that non-zero element. The set of positive integers under addition is at least a monoid.
Further structures become a bit easier to define. Creating a group requires that the set contain inverses under the operation. For the positive integers, it is impossible to add any element to any non-zero element to get zero as a sum. But if our set contained all integers, positive and negative, this would be possible. Thus the set of all integers under addition is at least a group.
If the group is commutative under the operation, it is called an Abelian group. Addition is commutative, so the set of all integers is at least an Abelian group. By extension, any group that is not commutative under the operation is non-Abelian.
Much work can be done on groups with just these few simple properties. This subfield of study is known as group theory.
A ring is a set of elements that is an Abelian group under one binary operation and a monoid under another. In our example, we have already shown that the set of all integers is an Abelian group under addition. If we create the multiplication operation, we see that the properties of a monoid (closure, associate binary operation, existence of identity element) all hold, so the set of all integers is at least a ring.
You can see that our example is a bit more powerful than that. Integers happen to be commutative under multiplication, so we can call this a commutative ring.
At this point we must stop short with our example. We have seemingly exhausted the properties of the integers, but you will notice that we have not quite said that the set of all integers is both a group under addition and a group under multiplication. Integers do not contain multiplicative inverses. That is to say, there is no way to multiply any non-one (identity) elements together to produce one (the identity). We know a set which does satisfy this condition: the set of all rational numbers. We can therefore say that the set of all real numbers is a field.
Beyond these simple set definitions is an entire world of symmetries, morphisms and more, all of which are studied in the field of abstract algebra. For further information about this field of mathematics, check your favorite bookseller for introductory college level textbooks on the subject.
