Algebraic rings

Overview and basic definitions

Definition: Ring

A ring $(R,+,\cdot )$ is a set with two binary operations such that:

• (i) $(R,+)$ is an abelian group;
• (ii) $\cdot$ is an associative binary operation;
• (iii) Left and right distributive laws hold.

In this case, we write $0$ for the additive identity of $R$ and $-r$ for the additive inverse of each $r\in R$. If multiplication $\cdot$ is commutative, then $R$ is called a commutative ring.

Definition: Subring

A subring $S\le R$ is set satisfying the ring axioms via the following properties:

• (i) Subgroup under $+$: $(S,+)\le (R,+)$;
• (ii) Closure under multiplication: For all $s_1,s_2\in S$, we have $s_1{s}_2\in S$ as well.

Related: Algebraic fields, Ring homomorphisms and isomorphisms, Rings of functions

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Familiar operations on general rings

Many familiar operations with numbers carry over to general rings. It follows from the basic ring axioms that for all $r\in R$, we have:

• $r0=0r=0$, since $r\cdot 0=r(0+0)=r0+r0⟹r0=0,$ and similarly for $0r$.
• $(-r)s=-(rs)=r(-s)$, since $0=0s=(r+(-r))s=rs+-rs⟹(-r)s=-(rs).$
• Exponential relations: For $n\in \mathbb{Z}$, we have
  • $n\cdot r=r+\cdots +r$ if $n>0$;
  • $0\cdot r=0$;
  • $n\cdot r=-((-n)\cdot r)$ for $n<0$.
• Laws of exponents: For $n,m\in \mathbb{Z}$,
  • $(n+m)\cdot r=(n\cdot r)+(m\cdot r)$;
  • $n\cdot (m\cdot r)=(nm)\cdot r$.
• General distributive law: $\left({\sum }_{i=1}^n{r}_i\right)\left({\sum }_{j=1}^m{s}_j\right)={\sum }_{1\le i\le n,1\le j\le m}(r_i{s}_j).$
• Exponential operation for multiplication: For $n\in \mathbb{N}$, we have $r^n=r\cdots r$, the $n$-times product of $r$ with itself. In particular, $r^1=r{r}^{n+1}=r\cdot r^n$
• $(r+s{)}^2=(r+s)(r+s)=r^2+sr+rs+s^2.$ If $R$ is commutative, then the following also hold:
  • $(r+s{)}^2=r^2+2rs+s^2$;
  • Binomial theorem: $(r+s{)}^n={\sum }_{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right)\cdot r^{n-i}{s}^i,$ where the end terms are defined by $r^n{s}^0=r^n$ and $r^0{s}^n=s^n$.

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Inverses and rings with unity

Definition: Rings with unity

We say $R$ is a ring with unity if there exists a unique multiplicative identity $1\in R$, called the unity, such that for all $r\in R$ we have $1r=r1=r$.

Further, if $R$ is a ring with unity, we can define $r^0=1$. Given $n\in \mathbb{Z}$, we also have $n\cdot 1\in R$ such that $(n\cdot 1)r=n\cdot (1r)=n\cdot r.$

If $R$ is a ring with unity, then $R^{\ast }$ is the set of units of $R$, and $(R^{\ast },\cdot )$ is a group.

Definition: Unit, invertible

Let $R$ be a ring with unity. An element $r\in R$ is a unit (or called invertible) if it has a multiplicative inverse, meaning there exists a unique element $r’\in \mathbb{R}$ such that $r{r}^{\prime }=r^{\prime }r=1.$

If $r,s$ are units, then so is $rs$. If $r\in R$ is invertible, we denote its unique inverse by $r^{-1}$, and if $r$ is a unit then so is $r^{-1}$ with $(r^{-1}{)}^{-1}=r$.

Lemma ( Modern Algebra II HW 2.1): Inverses are preserved under homomorphisms

Let $R,S$ be rings with unity and $f:R\to S$ be a ring homomorphism. If $r\in R^{\ast }$ is a unit in $R$, then $f(r)\in S^{\ast }$ is a unit in $S$ and

$$f(r{)}^{-1}=f(r^{-1}).$$

• Converse is not true

• Division ring, can add, subtract, multiply, and divide but not by 0

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Examples

• Integers modulo n
• Quaternions
• Polynomial rings
• Gaussian integer
• Rings of functions

The zero ring and abelian groups

The zero ring $\{0\}$ is defined to be the commutative ring where the additive and multiplicative identity are equal. It has the unique binary operations

$$0+0=00\cdot 0=0.$$

Notice also that $\{0\}\le R$ is a subring for any other ring $R$.

More generally, if $A$ is any abelian group, then $A$ is a commutative ring when equipped with multiplication defined by $a\cdot b=0$ for all $a,b\in A$; however, $A$ is a commutative ring with unity iff $A$ is the zero ring.

Square matrices of rings

Let $M_n(\mathbb{R})$ be the set of $n\times n$ matrices with coefficients in $\mathbb{R}$. This is a ring by the usual operations of addition and multiplication (multiplication is associative, left and right distributive laws hold); note that $M_n(\mathbb{R})$ is not commutative for $n>1$. We also have

$$I\in M_n(\mathbb{R})(M_n(\mathbb{R}){)}^{\ast }=G{L}_n(\mathbb{R}).$$

The same holds for matrices in $\mathbb{Z},\mathbb{Q},\mathbb{C}$, etc.

More generally, if $R$ is any ring, then $M_n(R)$ is as well. For example, $M_n(\mathbb{Z}/k\mathbb{Z})$ (note, again, that this does not commute for $n>1$) is a finite ring with $k^{n^2}$ elements.

Cartesian products

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