Changes

==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.

*For any ring <math>R</math> the polynomial ring <math>R[x]</math> consisting of elements of the form <math>b_0 +b_1x+...+b_mx^m</math> where <math>b_i \in R</math> is a Ring as well. All polynomials with integer coefficients form the ring <math>Z[x]</math>

*For any ring <math>R</math> the polynomial ring <math>R[x]</math> consisting of elements of the form <math>b_0 +b_1x+...+b_mx^m</math> where <math>b_i \in R</math> is a ring as well. All polynomials with integer coefficients form the ring <math>Z[x]</math>.

*<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! This means that this ring is not an [[domain]].

*<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.

*<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.

==Axioms==

==Axioms==