Introduction

A ring is a nonemtpy Set $R$ together with two Binary operations:

Addition. $+: R \times R \to R$. $(a,b) \mapsto a+b$

Multiplication. $\cdot : R \times R \to R$. $(a,b) \mapsto a \cdot b$

and an element $0 \in R$, such that

• $(R,+)$ is a commutative group with neutral element $0$ (like the Integers with addition, for e.g.). We use the same notation as for standard Addition.
• $\cdot$ is associative. $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Distributive law (compatibility): $a \cdot (b+c) = a\cdot b + a \cdot c$ and $(b+c) \cdot = b \cdot a + c \cdot a$

$(R,+)$ is called the Additive group of $R$. $0$ is also called the zero element of $R$. For notation, we can also omit $\cdot$, and it is implied.

Example: $(\mathbb{Z}, +, \cdot)$ (the Integers).

We can define Subring

Commutative ring

A Unit element is the Neutral element of multiplication, which is unique. A ring which contains a unit element is called a Unital ring or ring with 1.

Example: real 2x2 matrices.

Another example. Modular arithmetic

Remarks:

• $a0 = 0 = 0a$
• $(-a)b = a(-b) = -(ab)$
• $(-a)(-b)=ab$
• If R is a ring with 1, $(-1)a=a(-1)=-a$

Integral domain

Ring unit

can define the exponential (Exponential)

Polynomial ring

Division ring

Field

intro from Cryptography lectures.

multiplicative inverse