# Ring Theory/Rings

We will start by the definition of a ring.

Definition 1: A ring is a non empty set R together with two binary compositions defined by + and ., and satisfying the following properties hold for any ${\displaystyle a,b,c\in R}$:

• ${\displaystyle a+b\in R}$
• ${\displaystyle a+b=b+a}$
• ${\displaystyle a+(b+c)=(a+b)+c}$
• There exists an element denoted by ${\displaystyle 0\in R}$ such that ${\displaystyle a+0=a}$. 0 is called the additive identity or the zero element in R.
• For each ${\displaystyle a\in R}$, there exists an element ${\displaystyle b\in R}$ such that ${\displaystyle a+b=0}$. b is called additive inverse or negative of a and is written as b=-a so that a+(-a)=0.
• ${\displaystyle a.b\in R}$
• ${\displaystyle a.(b.c)=(a.b).c}$
• ${\displaystyle a.(b+c)=a.b+a.c}$ (Left distributive law.)
• ${\displaystyle (a+b).c=a.c+b.c}$ (Right distributive law.)

We denote a ring by (R,+,.). When the context is clear we just talk about a ring R and assume that the operations + and . are implicit. We will also drop the . in the operation a.b and just say ab.

The first 5 axioms of a ring just mean that (R,+) is an abelian group. The next two mean that (R,.) is a semi group. A ring is called commutative if ${\displaystyle a.b=b.a\ \forall a,b\in R}$. A ring is called boolean if ${\displaystyle x^{2}=x\ \forall x\in R}$. A ring R is called a ring with a unit element or unity or identity if ${\displaystyle \exists }$ an element ${\displaystyle e\in R}$ such that ${\displaystyle ae=ea=a\ \forall a\in R}$. Let R be a ring with unit element e. An element ${\displaystyle a\in R}$ is called invertible, if there exists an element ${\displaystyle b\in R}$ such that ${\displaystyle ab=ba=e}$. If n is a positive integer and a an element of a ring R then we define ${\displaystyle a^{n}=\underbrace {aa\cdots a} _{n\ times}}$ and ${\displaystyle na=\underbrace {a+a\cdots +a} _{n\ times}}$.

## Examples

One of the most important rings is the ring of integers ${\displaystyle \mathbb {Z} }$ with usual addition and multiplication playing the roles of + and . respectively. It is a commutative ring with identity as 1. The set of even numbers ${\displaystyle 2\mathbb {Z} :=\{0,\pm 2,\pm 4\cdots \}}$ is an example of a ring without identity. Like ${\displaystyle \mathbb {Z} }$, the sets of rational numbers ${\displaystyle \mathbb {Q} }$, of real numbers ${\displaystyle \mathbb {R} }$ and of complex numbers ${\displaystyle \mathbb {C} }$ are also rings with identity. However ${\displaystyle \mathbb {N} }$ is not a ring.

The ring of Gaussian integers is given by the set ${\displaystyle \mathbb {Z} [i]=\{m+ni:m,n\in \mathbb {Z} \}}$ where usual addition and multiplication of complex numbers are the operations. Here i stands (0,1) as is usual in the complex plane.

The set of all n by n matrices with real entries is an example of a non commutative ring with identity, under the usual addition and multiplication of matrices.

### The ring of integers modulo n

We now digress slightly to discuss a special kind of an equivalence relation which gives rise to an important class of finite rings.

Let n be a positive integer. Two integers a and b are said to be congruent modulo n, if their difference a − b is an integer multiple of n. If this is the case, it is expressed as:

${\displaystyle a\equiv b{\pmod {n}}.\,}$

The above mathematical statement is read: "a is congruent to b modulo n".

For example,

${\displaystyle 38\equiv 14{\pmod {12}}\,}$

because 38 − 14 = 24, which is a multiple of 12. For positive n and non-negative a and b, congruence of a and b can also be thought of as asserting that these two numbers have the same remainder after dividing by the modulus n. So,

${\displaystyle 38\equiv 2{\pmod {12}}\,}$

because both numbers, when divided by 12, have the same remainder (2). Equivalently, the fractional parts of doing a full division of each of the numbers by 12 are the same: .1666... (38/12 = 3.166..., 2/12 = .1666...). From the prior definition we also see that their difference, a - b = 36, is a whole number (integer) multiple of 12 ( n = 12, 36/12 = 3).

The same rule holds for negative values of a:

${\displaystyle -3\equiv 2{\pmod {5}}.\,}$

The properties that make this relation a congruence relation (respecting addition, subtraction, and multiplication) are the following.

If ${\displaystyle a_{1}\equiv b_{1}{\pmod {n}}}$ and ${\displaystyle a_{2}\equiv b_{2}{\pmod {n}}}$, then:

• ${\displaystyle (a_{1}+a_{2})\equiv (b_{1}+b_{2}){\pmod {n}}\,}$
• ${\displaystyle (a_{1}-a_{2})\equiv (b_{1}-b_{2}){\pmod {n}}\,}$
• ${\displaystyle (a_{1}a_{2})\equiv (b_{1}b_{2}){\pmod {n}}.\,}$

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by ${\displaystyle {\overline {a}}_{n}}$, is the set ${\displaystyle \left\{\ldots ,a-2n,a-n,a,a+n,a+2n,\ldots \right\}}$. This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class of a modulo n. Another notation for this congruence class, which requires that in the context the modulus is known, is ${\displaystyle \displaystyle [a]}$.

The set of congruence classes modulo n is denoted as ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ (or, alternatively, ${\displaystyle \mathbb {Z} /n}$ or ${\displaystyle \mathbb {Z} _{n}}$) and defined by:

${\displaystyle \mathbb {Z} /n\mathbb {Z} =\left\{{\overline {a}}_{n}|a\in \mathbb {Z} \right\}.}$

When n ≠ 0, ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ has n elements, and can be written as:

${\displaystyle \mathbb {Z} /n\mathbb {Z} =\left\{{\overline {0}}_{n},{\overline {1}}_{n},{\overline {2}}_{n},\ldots ,{\overline {n-1}}_{n}\right\}.}$

We can define addition, subtraction, and multiplication on ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ by the following rules:

• ${\displaystyle {\overline {a}}_{n}+{\overline {b}}_{n}={\overline {a+b}}_{n}}$
• ${\displaystyle {\overline {a}}_{n}-{\overline {b}}_{n}={\overline {a-b}}_{n}}$
• ${\displaystyle {\overline {a}}_{n}{\overline {b}}_{n}={\overline {ab}}_{n}.}$

The verification that this is a proper definition uses the properties given before.

In this way, ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ becomes a commutative ring. For example, in the ring ${\displaystyle \mathbb {Z} /24\mathbb {Z} }$, we have

${\displaystyle {\overline {12}}_{24}+{\overline {21}}_{24}={\overline {9}}_{24}}$

as in the arithmetic for the 24-hour clock.