Integer ring

Time:2021-1-17

ring

Integer ringZ_mIt consists of the following two parts:

1. AssemblyZ_m = ( 0,1,2,…,m )

2. Two kinds of operations “+” and “X” make thea, b\in Z_myes:

a + b\equiv c\ mod\ m, (c\in Z_m)\\
a\times b\equiv d\ mod\ m, (d\in Z_m)

Key features of rings

  • If the result of the addition or multiplication of any two numbers in a ring is always in the ring, then the ring is closed.
  • Addition and multiplication can be combined,

For example, for alla,b,c\in Z_m, have $$a + (B + C) = (a + b) + C and a · (B · C) = (a · b) · C

  • There is a neutral element 0 in addition,

So that for eacha\in Z_mAll of thema + 0\equiv a\ mod\ m

  • Any element a in a ring has a negative element – A,

bringa + ( -a )\equiv 0\ mod\ mThat is, the additive inverse always exists.

  • There is a neutral element 1 in multiplication,

That is, for eacha\in Z_mAll of thema\times 1\equiv a\ mod\ m

  • Not all elements have multiplicative inverses.

hypothesisa\in ZThe inverse of multiplicationa^{-1}It can be defined as:

a·a^{-1}\equiv 1\ mod\ m

If the multiplicative inverse of element a exists, it can be divided by this element becauseb/a\equiv b·a^{-1}\ mod\ m

  • It is difficult to find the inverse of an element (usually Euclidean algorithm is used), but a simple method can be used to determine whether the inverse of a given element a exists

If and only if GCD (a, m) = 1, an elementa\in ZExistence of multiplicative inversea^{-1}. GCD is the greatest common divisor, which is the largest integer that can divide a and m at the same time. In number theory, it is of great significance that the greatest common divisor of two numbers is 1, and it has a special name, that is: GCD (a, m) = 1, then a and m are called coprime or coprime.

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