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Line 5:
Line 5: − *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".
As I've restructured this page to use the definition of a ring with idenity, I've removed the non-ring example (even integers which is a rng) and replaced it with the polynomial ring
*<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!
*<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.