A ring is a circular band of gold, silver, or other precious or decorative material worn on the finger (see also earring and necklace). Rings have long been used as ornaments and symbol of love, fidelity, friendship, and authority. They have been a key element in the evolution of culture, history, and religion. They are sometimes worn on other fingers, the toes, or through the nose.
A group RG is said to be simple if its ring RR has no two-sided ideals, or more generally if its ring has no nontrivial subgroups. This definition is often used in algebra and other fields to distinguish groups that are easy to study from those that are more difficult. For example, the ring of integers is a simple ring. On the other hand, the ring of real square matrices is not a simple ring because it has multiple nontrivial two-sided ideals.
The simplest rings are those which satisfy the commutativity of addition and multiplication. These are called commutative rings. They include the ring of integers, the ring of polynomials in one and two variables, and the ring of real square matrices. The ring of integers is the simplest commutative ring with a unit element, and it has an additive identity and multiplicative inverse, as well as being closed under composition.
Another way to view the ring concept is as an object of higher category theory. Any preadditive category can be considered to have a ring object, and many of the definitions and theorems originally given for rings are valid for these objects. Furthermore, the notion of ring homomorphism can be generalized to these categories by considering the homomorphisms that “factor through” right- or left-multiplication.
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