Ring (mathematics) - Basic Examples

Basic Examples

Commutative rings:

• The motivating example is the ring of integers with the two operations of addition and multiplication. This is a commutative ring.
• The rational, real and complex numbers form commutative rings (in fact, they are even fields).
• The Gaussian integers form a ring, as do the Eisenstein integers. So does their generalization Kummer ring. cf. quadratic integers.
• In general, the set of all algebraic integers forms a ring. This follows for example from the fact that it is the integral closure of the ring of rational integers in the field of complex numbers. The rings in the previous example are subrings of this ring.
• The polynomial ring R of polynomials over a ring R is also a ring.
• The set of formal power series R] over a commutative ring R is a ring.
• If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This corresponds to a ring of sets and is an example of a Boolean ring.
• The set of all continuous real-valued functions defined on the real line forms a commutative ring. The operations are addition and multiplication of functions.
• Let R be the set of all continuous functions on the real line that vanish outside a bounded interval (an interval depends on a function). One can consider the following multiplication:

R is then a ring but not unital: this is because if there were the multiplicative identity, it must be Dirac's delta function, which is not a continuous function.

Noncommutative rings:

• For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n=1, this matrix ring is just (isomorphic to) R itself. For n>1, this matrix ring is an example of a noncommutative ring.
• If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms.
• If G is a group and R is a ring, the group ring of G over R is a free module over R having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
• Many rings that appear in analysis are noncommutative. A basic example is a Banach algebra.

Non-example:

• The set of natural numbers N is not a ring, since (N, +) is not even a group (the elements are not all invertible with respect to addition). For instance, there is no natural number which can be added to 3 to get 0 as a result. There is a natural way to make it a ring by adding negative numbers to the set, thus obtaining the ring of integers. The natural numbers (including 0) form an algebraic structure known as a semiring (which has all of the properties of a ring except the additive inverse property).