Abstract algebra/Rings, fields and modules

Introduction to Rings

Rings are algebraic structures designed to model and abstract the structure of the integers ($\mathbb{Z}$), so that we can duplicate some of the processes in which integers are used, but in a more general setting. It will be helpful if you have familiarity with the concepts and theorems for groups, because we'll be using many of the same ideas and theorems.

Definition: A ring is a set $R$ with two binary operations $+$ and $\cdot$ that satisfies the following properties:

For all $a,b,c\in R,$

1. $a+b\in R$ ($R$ is closed under $+$)
2. $(a+b)+c=a+(b+c)$ ($+$ is associative)
3. $\mathrm{\exists }0\in R:0+a=a$ ($R$ contains an additive identitiy)
4. $\mathrm{\exists }-a\in R:a+(-a)=0$ ($R$ contains additive inverses)
5. $a+b=b+a$ ($+$ is commutative)
6. $a\cdot b\in R$ ($R$ is closed under $\cdot$)
7. $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ ($\cdot$ is associative)
8. $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ and $(b+c)\cdot a=(b\cdot a)+(c\cdot a)$ ($\cdot$ is distributive over $+$)

If you're already familiar with the concepts of groups and semigroups, we can compress the conditions above to:

1. $(R,+)$ is an abelian group
2. $(R,\cdot )$ is a semigroup
3. $\cdot$ is distributive over $+$.

We'll often use juxtaposition in place of $\cdot$, i.e., $ab$ for $a\cdot b$.

Note that a ring does not necessarily have the property of commutative multiplication. When it does, the ring is called a commutative ring. Also, a ring does not necessarily have a multiplicative identity. A nonzero element of a ring that is an identity under multiplication is sometimes called a unity.

Examples:

1. The set $\mathbb{Z}$ of integers under addition and multiplication is a commutative ring with unity 1.
2. The set $2\mathbb{Z}$ of even integers under addition and multiplication is a commutative ring without unity.
3. The set $M_2$ of 2x2 matrices composed of integers is a noncommutative ring with unity $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$.

Theorem: If a ring has a unity, that unity is unique.

Proof: Let $R$ be a ring with unity $u$. Now suppose $x$ is another unity in $R$. Then $x=x\cdot u=u$, and $x=u$.

Of course proving that a set with two operations satisfy all of the above conditions can be tedious. So, just as we did for groups, we note that if we're considering a subset of something that's already a ring, then our job is easier.

Definition: A subring $S$ of a ring $R$ is a subset of $R$ that is a ring (under the same two operations as for $R$).

Theorem: If $S$ is a subset of a ring $R$, then $S$ is a subring of $R$ if:

1. $S$ is nonempty
2. $S$ is closed under $-$
3. $S$ is closed under $\cdot$

Examples:

1. The trivial subring $\left\{0\right\}$ is a subring of every ring.
2. The set of Gaussian integers $Z\left[i\right]=\{a+bi|a,b\in Z\}$ is a subring of the complex numbers $C$.

Types of Commutative Rings

Definition: Let $R$ be a commutative ring with a $1$. An element $r\in R$ is a unit if there is an element $r^{-1}\in R$ such that $r{r}^{-1}=1$. The set of all units is denoted by $R^{\star }$ and is a group under the multiplication operation.

Definition: We say an element $r\in R$ is a zero-divisor if there is an element $s\in R$ such that rs = 0.

Observe that a zero-divisor may not be a unit (unless 1 = 0 in which case R = 0).

Definition: We say a commutative ring $R$ with a $1\ne 0$ is an integral domain if the only zero-divisor is $0$

Definition: We say a commutative ring $R$ with a $1\ne 0$ is a field if all the non-zero elements are units.