| 单词 | ring |
| 释义 | ring (i) for all a, b, and c in R, a+(b + c) = (a + b) + c, (ii) for all a and b in R, a + b = b + a, (iii) there is an element 0 in R such that a+0 = a for all a in R, (iv) for each a in R, there is an element −a in R such that a+(−a) = 0, (v) for all a, b, and c in R, a(bc) = (ab)c, (vi) for all a, b, and c in R, a(b + c) = ab + ac and (a + b)c = ac + bc. The element 0 guaranteed by (iii) is an additive identity. It can be shown to be unique and has the extra property that a0 = 0 for all a in R; it is called zero. Also, for each a, the element −a guaranteed by (iv) is unique and is the negative or additive inverse of a. The ring is a commutative ring if it is further true that (vii) for all a and b in R, ab = ba,and it is a commutative ring with identity if also (viii) there is an element 1 (≠0) such that a1 = a for all a in R. The element 1 guaranteed by (viii) is a multiplicative identity. It can be shown to be unique and is referred to as ‘one'. Further properties may be required for other types of ring such as integral domains and fields. Examples of rings include the set of 2 × 2 real matrices and the set of all even integers, each with the appropriate addition and multiplication. Another example is ℤn, the set of integers with addition and multiplication modulo n. A ring may be denoted by (R,+,×) when it is necessary to be clear about the ring's operations. But it is sufficient to refer simply to the ring R when the operations intended are clear. |
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