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'''1.''' ''R'' with addition is a [[commutative]] [[group]];  

'''1.''' ''R'' with addition is a [[commutative]] [[Group (mathematics)|group]];  

'''2.''' ''R'' is closed under multiplication;

'''2.''' ''R'' is closed under multiplication;

==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.

*<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!

A '''commutative ring''' is a ring in which multiplication is commutative.

A '''commutative ring''' is a ring in which multiplication is commutative.

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

A '''division ring''' is a ring such that ''R'' with multiplication is a (not necessarily [[commutative]]) [[Group (mathematics)|group]].

[[Category:Algebra]]

[[Category:Algebra]]