Ring

Definition of a Ring

Definition 5.18

A ring $〈R;+,-,0,\cdot ,1〉$ is an algebra for which (i) $〈R;+,-,0〉$ is a commutative group. (ii) $〈R;\cdot ,1〉$ is a Monoid. (iii) $a(b+c)=(ab)+(ac)$ and $(b+c)a=(ba)+(ca)$ for all $a,b,c\in R$ (left and right distributive laws).

[[Math/ Books/A First Course in Abstract Algebra.pdf#page=172&selection=70,0,210,1&color=note|p.167]]

A ring $⟨\mathbb{R},+,\cdot ⟩$ is a set $R$ together with two binary operations + and ·, which we call addition and multiplication, defined on $R$ such that the following axioms are satisfied:

• R1. $⟨\mathbb{R},+,\cdot ⟩$ is an abelian group.
• R2. Multiplication is associative.
• R3. For all a, b, c ∈ R, the left distributive law, a · (b + c) = (a · b) + (a · c) and the right distributive law (a + b) · c = (a · c) + (b · c) hold.

NOTE

A ring is called commutative if multiplication is commutative (ab = ba)

Examples of Rings

1) complex numbers

$R^{\prime }=⟨R\times R;+,-,(0,0),\ast ,(1,0)⟩$

• $(a,b)+(c,d)\stackrel{def}{=}(a+c,b+d)$
• $(a,b)\ast (c,d)\stackrel{def}{=}(ac-bd,ad+bc)$

If $R=\mathbb{R}$ then $R^{\prime }$ is actually $\mathbb{C}$

2) polynomial ring

$R^{\prime \prime }=⟨R^{\ast };+,-,\epsilon ,\ast ,(1)⟩$ (Note that $R^{\ast }$ means here the finite sequence of $R$)

• $(f_0,\dots ,f_n)+(g_0,\dots ,g_m)\stackrel{def}{=}(f_0+g_0,f_1+g_1,\dots ,f_n+g_n,g_{n+1},\dots ,g_m)$ $(n\le m)$ (extend the shorter tuple by zeros)
• $(f_0,\dots ,f_n)\ast (g_0,\dots ,g_m)\stackrel{def}{=}(f_0{g}_0,f_1{g}_0+f_0{g}_1,f_2{g}_0+f_1{g}_1+f_0{g}_2,\dots )$

This is the polynomial ring in one indeterminate over $R$: $R^{\prime \prime }\simeq R[x]$ $(1)$ is the constant polynomial $1$

Lemma 5.17: Properties of a ring

Lemma 5.17

i) $0\cdot a=a\cdot 0=0$ ii) $(-a)b=-(ab)=a(-b)$ iii) $(-a)(-b)=ab$ iv) $|R|=1⟺1=0$ (If $R$ is non-trivial, then $1\ne 0$)

Proof iv) Prove $⟸$: trivial Prove $⟹$: Let $1=0$, assume $|R|>1$ $a\in R,a\ne 0$ $⟹0=a\cdot 0=a\cdot 1=a⟹$ Contradiction $⟹|R|=1$

Definition: Characteristic of a ring

(DE: Charakteristik) The characteristic of a ring is the order of $1$ in the additive group if it is finite, and otherwise the characteristic is defined to be $0$ Formally:

$$min\{i\in \mathbb{N}|i>0\wedge \underset{iMal}{1+1+\cdots +1}=0\}\vee 0$$

[[Section 19 Integral Domains#1913-definition-the-characteristic-of-the-ring-r|19.13 Definition The Characteristic of the ring $R$]]

Definition: Unit

(DE: Einheit) An element $u$ of a ring $R$ is called a unit if $u$ is invertible That is $u{u}^{-1}=u^{-1}u=1$ for some $u^{-1}\in R$ The set of units is denoted by $R^{\ast }$ (must be a group)

Example

The units of $\mathbb{Z}$ are $-1$ and $1$, ${\mathbb{Z}}^{\ast }=\{-1,1\}$

Lemma 5.18: Multiplicative Group of Units

Lemma 5.18

$⟨R^{\ast },\cdot ,(\cdot {)}^{-1},1⟩$ is a group

Definition: Zero Divisor

$a\in R\setminus \{0\}$ is zero divisor, if there exists $b\in R\setminus \{0\}$ such that $ab=0$

Applications

• Error-Correcting Codes: Rings are used in the construction of error-correcting codes, which are essential for reliable data transmission and storage.
• Secret Sharing: Rings play a role in secret sharing schemes, where a secret is divided into multiple shares distributed among participants, and a certain number of shares are required to reconstruct the secret.
• Privacy Amplification: Rings are used in privacy amplification techniques, where a partially secret shared value (e.g., a bitstring) can be transformed into a shorter, more secret value. A public function f compresses the bitstring (e.g., from 1000 bits to 800 bits). The specific choice of f is public knowledge, but the way it is applied depends on an additional secret. The remaining 200 bits could function like this key, akin to a one-time pad.