Difference between revisions of "Ring (mathematics)"
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==Examples== | ==Examples== | ||
*<math>(\mathbb{Z},+, \cdot)</math> - the set of the [[integers]] - together with the usual [[addition]] and [[multiplication]] is a ring. | *<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. | + | *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>(\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== | ==Remarks== | ||
A '''ring with unity''' is a ring for which multiplication has a [[neutral element]]. | 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. | + | 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 (mathematics)|group]]. | A '''division ring''' is a ring such that ''R'' with multiplication is a (not necessarily [[commutative]]) [[Group (mathematics)|group]]. | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Revision as of 22:00, June 13, 2009
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
- - 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".
- : 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, , so, you can multiply two elements, neither of which is zero, and get zero as the result!
- , the set of 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.