# 1. Rings
# 1.1 Definition 1.4.2 Ring
The integer ring Zm consists of:
- The set Zm = {0,1,2, . . . ,m−1}
- Two operations “+” and “×” for all a,b ∈ Zm such that:
- a+b ≡ c mod m , (c ∈ Zm)
- a×b ≡ d mod m , (d ∈ Zm)
e.g. Z9 ={0,1,2,3,4,5,6,7,8}.
Note:
- for Additive Composition 加法环是abelian group，
- but for Multiplicative Composition 而乘法环不一定
Properties of rings：
In summary, we can say that the ring Zm is the set of integers {0,1,2, . . . ,m−1} in which we can add, subtract, multiply, and sometimes divide.
# 1.2 Property: Closed(addition&multiplication)
We can add and multiply any two numbers and the result is always in the ring. A ring is said to be closed.
# 1.3 Property: Associative(addition&multiplication)
Addition and multiplication are associative, e.g., a+(b+c) = (a+b)+c, and a · (b · c) = (a · b) · c for all a,b,c ∈ Zm.
# 1.4 Property: Additive Identity/Neutral element 0
There is the neutral element 0 with respect to addition, i.e., for every element a ∈ Zm it holds that a+0 ≡ a mod m.
# 1.5 Property: Additive Inverse(Uniqueness)
For any element a in the ring, there is always the negative element −a such that a+(−a) ≡0 mod m, i.e., the additive inverse always exists.
唯一性，反证利用结合律：
对于x ∈ Zm 如果存在两个不同的加法逆元a,b ∈ Zm，st. x+a=a+x=x+b=b+x=0，
根据结合律 a+x+b = (a+x)+b = 0+b=b , a+x+b=a+(x+b)=a+0=a，跟a和b不同矛盾
负数的模： mod(a, n) = a - n * floor(a / n)
# 1.6 Property: Additive Communitativity
for all a,b ∈ Zm, a+b=b+a
# 1.7 Property: distributive(Connection between addition&multiplication)
Distributive law holds that a×(b+c) = (a×b)+(a×c) for all a,b,c ∈ Zm,
Expand 扩展：
(x1+x2+....xm)×(y1+y2+...yn) = ..
可以将 (x1+x2+....xm)先看做a，然后b=y1, c=y2+...yn，然后展开即可
# 1.8 Property: Multiplicative Identity/Neutral element 1
There is the neutral element 1 with respect to multiplication, i.e., for every element a ∈ Zm it holds that a×1 ≡ a mod m.
# 1.9 Property: Multiplicative Communitativity(NA)
# 1.10 Property: Multiplicative Inverse(NA)
The multiplicative inverse exists only for some, but not for all, elements. Let a ∈ Z, the inverse a−1 is defined such that a · a−1 ≡1 mod m If an inverse exists for a, we can divide by this element since b/a ≡ b · a−1 mod m.
It takes some effort to find the inverse (usually employing the Euclidean algorithm). However, there is an easy way of telling whether an inverse for a given element a exists or not: An element a ∈ Z has a multiplicative inverse a−1 if and only if gcd(a,m) = 1, where gcd is the greatest common divisor , i.e., the largest integer that divides both numbers a and m. The fact that two numbers have a gcd of 1 is of great importance in number theory, and there is a special name for it: if gcd(a,m) = 1, then a and m are said to be relatively prime or coprime 互质.
# 如果存在multiplicative Inverse,则有如下几个性质:
唯一性
跟前面additive inverse一样反证利用结合律：
对于x ∈ Zm 如果存在两个不同的加法逆元a,b ∈ Zm，st. xa=ax=xb=bx=1，
根据结合律 axb = (ax)b = 1b=b , axb=a(xb)=a1=a，跟a和b不同矛盾
x对n模运算的乘法逆存在的前提是公约数gcd(x,n)=1
证明Proving that modular inverse only exists when gcd(n,x)=1https://math.stackexchange.com/questions/2101189/proving-that-modular-inverse-only-exists-when-gcdn-x-1
If there is an inverse of x mod n, that gives us a number y so that xy≡1mod n . That means that xy=kn+1, or (rearranging) that xy−kn=1.
Now for any common divisor, c, of x and n we will have that c∣(xy−kn) which gives c∣1, that is, c=1. So that is an outcome - and therefore a requirement - of finding the inverse of x mod n
# Unit
x ∈ R ,x is unit if ∃ 1/x = x-1 ∈ R
s.t. x(1/x)=1 , (1/x)x=1
# 2. Communitative Rings
based on the definition of Rings, just add one more axioms, that it must obey :
∀ x,y ∈ R , xy=yx
example:
Rational numbers,
Real numbers,
Complex numbers,
all the prime fields on vector space,
Integer Rings,
Zero Ring:
{0} the additive identity&multiplicative identity is the same 0
# 3. Sub Ring
Two trivial example:
- Zero Ring is sub ring of any Ring;
- any Ring is the sub ring of itself
# 4. Ring of Polynomials
R[x] = Ring of polynomials in the indeterminant x over the ring R
x0, x1, x2, x3, ...........
$$ \sum_{i=0}^∞ ai \quad $$