A binary relation \(\ast\) on a set \(S\) is some function that takes a pair \((a,b)\) for \(a,b \in S\) and returns some other member of \(S\) denoted \(a\ast b.\) A binary relation is associative if \(a\ast(b\ast c) = (a\ast b)\ast c\) for all \(a,b,c \in S.\) A binary relation is commutative if \(a\ast b = b\ast a\) for all \(a,b \in S.\) If there exists an member \(e \in S\) such that \(e \ast a = a = a \ast e\) for all \(a \in S,\) that member is called the identity. For \(a \in S\) if there exists another member \(b \in S\) such that \(a\ast b = e,\) then \(b\) is called a right-inverse of \(a.\) Imagine how a left-inverse is defined. If \(a\in S\) has a left- and right-inverse and they’re the same member, we call it the inverse of \(a\) and denote it \(a^{-1}.\)
A group \(G\) is a set equipped with an associative binary operation \(\ast\) that has an identity member, and for which ever member has an inverse. If the binary operation \(\ast\) on \(G\) is commutative too, we conventionally call \(G\) an abelian group, and sometime use the symbols “\(+\)” instead of “\(\ast\)”. If it’s clear from the context that \(G\) is a group, the \(\ast\) will often be omitted, and we’ll write \(a \ast b\) simply as \(ab.\) Every group can be thought of as collection of composable transformations or symmetries of a space; e.g. the permutations of a set, the isometries of the plane, the states of a Rubik’s cube, etc. Concretely for a group \(G\) with member/transformations \(a\) and \(b,\) \(ab\) denotes the composite transformation, usually the result of doing \(b,\) then \(a.\) As a matter of notation, we’ll denote the result of iteratively applying an transformation \(a\) repeatedly \(n\) times as \(a^n.\) The order of a member \(a\) of a group is the smallest \(n\) for which \(a^n = e.\)
A ring \(R\) is a set equipped with two binary operations, \(+\) and \(\ast,\) such that the operation \(+\) confers an abelian group structure on \(R,\) the operation \(\ast\) is associative, and together they obey the left- and right-distributive laws \({a\ast(b+c) = (a\ast b)+(a \ast c)}\) and \({(a+b)\ast c = (a\ast c)+(b \ast c).}\) A ring can be said to be commutative and have an identity (denoted \(1\)) and its member said to have inverses, all in reference to the operation \(\ast.\) The terminology with rings can differ though: if a ring has an identity then it’s usually called a “ring with unity.” Some folks insist that all rings should have an identity and will consider a ring without one as a degenerate ring and call it a rng. Finally, a field — like \(\mathbf{R}\) or \(\mathbf{C}\) — is a commutative ring with unity in which every member is invertible.