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added an example of a non-commutative ring!
==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]]