Ring (mathematics)

ID: ring-mathematics

In mathematics, specifically in abstract algebra, a **ring** is a set equipped with two binary operations that generalize the arithmetic of integers. Specifically, a ring consists of a set \( R \) together with two operations: addition (+) and multiplication (·). The structure must satisfy the following properties: 1. **Additive Closure**: For any \( a, b \in R \), the sum \( a + b \) is also in \( R \).
Ring (mathematics) by Ciro Santilli 37 Updated +Created
A Ring can be seen as a generalization of a field where:
Addition however has to be commutative and have inverses, i.e. it is an Abelian group.
The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.
A polynomial ring is another example with the same properties as the integers.
The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:
Note that is not a ring because you can by addition reach the zero matrix.

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