Changes

==Examples==

==Examples==

*<math>(\mathbb{Z},+, \dot)</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!