Rngs

Overview

A rng $⟨R,+,\cdot ⟩$ (pronounced "rung") is a set $R$ together with two binary operations $+$ and $\cdot$, called addition and multiplication respectively, defined on $R$ such that the following axioms are satisfied:

1. $⟨R,+⟩$ is an abelian group, called the additive group of the ring $R$.
2. Multiplication is associative.
   1. In other words, $⟨R,\cdot ⟩$ is a semigroup.
3. For all $a,b,c\in R$, the following hold:
   1. Left distributive law: $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$
   2. Right distributive law: $(a+b)\cdot c=(a\cdot c)+(b\cdot c)$

The additive identity element is denoted $0$. The additive inverse of an element $a$ is denoted $-a$. For some $n\in \mathbb{Z}$,

• $0a$ is interpreted as the additive identity element.
• $na$ is interpreted as $a+a+\cdots +a$, where $a$ is repeated $n$ times.
• $-na$ is interpreted as $(-a)+(-a)+\cdots +(-a)$, where $(-a)$ is repeated $n$ times.

A rng in which multiplication is commutative is a commutative rng.

Subrngs

Let $⟨R,+,\cdot ⟩$ be a rng. If $S\subseteq R$ is a rng under $+$ and $\cdot$, then $S$ is a subrng of $R$.

If $S$ is a subrng of $R$, then $⟨S,+⟩$ is a subgroup of $⟨R,+⟩$.

Criterion

A nonempty subset $S$ of a rng $R$ is a subrng if and only if

• $S$ is closed under subtraction, and
• $S$ is closed under multiplication.

Intersection

Let $\{S_i\mid i\in I\}$ be an indexed set of subrngs of $R$. Then their intersection $\bigcap _{i\in I}{S}_i$ is a subrng of $R$.

Homomorphisms

Let $R$ and $R^{\prime }$ be rngs. A rng homomorphism $\varphi :R\to R^{\prime }$ is a mapping that satisfies the following conditions for all $a,b\in R$:

1. $\varphi (a+b)=\varphi (a){+}^{\prime }\varphi (b)$
2. $\varphi (a\ast b)=\varphi (a)\ast \varphi (b)$

Note that $\varphi$ is also a group homomorphism so the linked structural properties are preserved. Additionally,

• Subrngs
  • If $S\le R$, then $\varphi [[S]]\le R^{\prime }$.
  • If $S^{\prime }\le R^{\prime }$, then ${\varphi }^{-1}[[S]]\le R$.

Direct Products

Let $R_1,R_2,\dots ,R_n$ be rngs. Let $a_i,b_i\in R_i$ for $1\le i\le n$ and define

Then the direct product $\prod _{i=1}^n{R}_i$ is a rng under these binary operations.

Characteristic

Let $R$ be a rng. If there exists an $n\in {\mathbb{Z}}^{+}$ such that $n\cdot a=0$ for all $a\in R$, then the least such positive integer, denoted $\text{char}(R)$, is the characteristic of $R$. If no such positive integer exists, then $R$ is of characteristic $0$.

If a rng has characteristic $0$, no information is provided about the characteristic of a subrng. For example, $\text{char}(\mathbb{Z}\times {\mathbb{Z}}_n)=0$ but $\text{char}(\{0\}\times {\mathbb{Z}}_n)=n$.

If a rng $R$ has characteristic $n\in {\mathbb{Z}}^{+}$, then a subrng $S\le R$ has characteristic $m\in {\mathbb{Z}}^{+}$ where $m\mid n$.

Elements

Idempotence

Let $⟨R,+,\cdot ⟩$ be a rng with element $a\in R$. Then $a$ is idempotent if $a^2=a$. If $a$ is idempotent, it inductively follows that $a^n=a$ for all $n\in {\mathbb{Z}}^{+}$.

Nilpotence

Let $⟨R,+,\cdot ⟩$ be a rng with element $a\in R$. Then $a$ is nilpotent if there exists some $n\in {\mathbb{Z}}^{+}$ such that $a^n=0$. The smallest such $n$ is called the index of nilpotency.

Divisors of 0

If $a$ and $b$ are two nonzero elements of a rng $R$ such that $ab=0$, then $a$ and $b$ are divisors of $0$ (or $0$ divisors).

The $0$ divisors of ${\mathbb{Z}}_n$ are precisely those nonzero elements that are not relatively prime to $n$.

Cancellation

Let $R$ be a rng and $a,b,c\in R$. The left and right cancellation laws respectively state that

These laws hold if and only if $R$ has no divisors of $0$.