Ring (mathematics)

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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

  • (\mathbb{Z},+, \cdot) - 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".
  • (\mathbb{Z} / 6\mathbb{Z},+, \cdot) : 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, 2 \cdot 3 = 0 , so, you can multiply two elements, neither of which is zero, and get zero as the result!
  • M^{n \times n}(\mathbb R), the set of n \times n 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.

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