Ring (mathematics)

A ring in mathematics is a set R equipped with two binary operations, usually called addition and multiplication, satisfying that

1. R with addition is a commutative group;

2. R is closed under multiplication;

3. Multiplication is associative;

4. Multiplication distributes over addition.

Examples

• <math>(\mathbb{Z},+, \cdot)</math> - the set of the integers - together with the usual addition and multiplication is a ring.
• the subring of the even numbers is a ring, too: this shows that there is not necessarily a neutral element of the multiplication in a ring. A ring without multiplicative identity is sometimes called (tongue-in-cheek) a "rng".
• <math>(\mathbb{Z} / 6\mathbb{Z},+, \cdot)</math> : this is the ring of six elements {0,1,2,3,4,5} and the usual addition and multiplication modulo six. So, here 1+3= 4, but 4+5 = 3. Interestingly, <math>2 \cdot 3 = 0 </math>, so, you can multiply two elements, neither of which is zero, and get zero as the result!
• <math>M^{n \times n}(\mathbb R)</math>, the set of <math>n \times n</math> real matrices, with operations of matrix addition and multiplication, is a ring.

Remarks

A ring with unity is a ring for which multiplication has a neutral element.

A commutative ring is a ring in which multiplication is commutative. The first three examples of rings given above are commutative rings: the last is not, since matrix multiplication is not in general commutative. The study of the properties of commutative rings is usually called commutative algebra.

A division ring is a ring such that R with multiplication is a (not necessarily commutative) group.