Abstract Algebra/Rings

Introduction to Rings

Rings are algebraic structures designed to model and abstract the structure of the integers ($\Bbb Z$), 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. (R, + ) is an abelian group:
   1. $a+b\in R$ (R is closed under + )
   2. $(a+b)+c=a+(b+c)\!$ ( + is associative)
   3. $\exists 0\in R : 0+a=a$ (R contains an additive identity)
   4. $\exists -a\in R : a+(-a)=0$ (R contains additive inverses)
   5. $a+b=b+a\!$ ( + is commutative)
2. $(R,\cdot)$ is a semigroup:
   1. $a\cdot b\in R$ (R is closed under $\cdot$)
   2. $(a\cdot b)\cdot c=a\cdot(b\cdot c)$ ($\cdot$ is associative)
3. $\cdot$ is distributive over + :
   1. $a\cdot(b+c)=(a\cdot b)+(a\cdot c)$
   2. $(b+c)\cdot a=(b\cdot a)+(c\cdot a)$

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

Examples

1. The set $\mathbb{Z}$ of integers under standard addition and multiplication.
2. The set $2\mathbb{Z}$ of even integers under standard addition and multiplication.
3. The sets $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ are also rings under standard addition and multiplication.
4. The set ${M}_2(\mathbb{R})$ of 2x2 square matrices with real coefficients under standard addition and multiplication.
5. The set $Maps(\mathbb{R},\mathbb{R})$ of functions on $\mathbb{R}$ with pointwise addition and multiplication.
6. More generally, if R is a ring, the set Maps(R,R) is also a ring.
7. The set Maps(R,R) with function composition for multiplication is not a ring since the statement $f\circ(g+h) = f\circ g + f\circ h$ is not true in general.
8. The set of integrable functions on the real numbers, L¹, is a ring under pointwise addition and multiplication given by convolution:

$(f*g)(t) = \int_\mathbb{R} f(\tau)g(t-\tau) d\tau$

This ring is important to the study of linear systems and differential equations.

1. The set {0} with the only possible binary operations is called the trivial ring or the zero ring. As we will see, all singleton sets are isomorphic to {0} when given operations. A ring which contains more than one element is called a non-zero ring.
2. The set of Gaussian integers $\mathbb{Z}\left[i\right]=\left\{a+bi|a,b\in\mathbb{Z}\right\}$ with standard addition and multiplication.
3. Let R be a ring, and let R[X] denote all the polynomials a_nXⁿ + a_{n − 1}X^{n − 1} + ... + a₁X + a₀ with coefficients in R. Then R[x] is a ring under standard addition and multiplication. If $f(x) = \sum_{i=0}^m a_ix^i$ and $g(x) = \sum_{i=0}^nb_ix^i$, then $(fg)(x) = \sum_{i=0}^{m+n}c_ix^i$ where $c_i = \sum_{j=0}^i a_jb_{i-j}$. Polynomial multiplication is also called discrete convolution.

Theorem: Let R be a ring, and let $a,b,c\in R$. Then the following are true:

1. If a + b = a + c, then b = c.
2. The equation a + x = b has a unique solution.
3. − ( − a) = a
4. 0a = 0
5. ( − a)b = − (ab)
6. ( − a)( − b) = ab

Proof: (1), (2), and (3) all strictly concern addition, and are all previous results from (R, + ) being a group. The other three parts all concern both addition and multiplication (since 0 and - are additive concepts), so as a proof strategy we expect to use the distributive law in some way to link the two operations. For (4), observe that 0a + 0a = (0 + 0)a = 0a. But then by (1), a=0. For (5), Note that (-a)b + ab = (-a + a)b = 0b = 0. For (6) note that (-a)(-b) + -(ab) = (-a)(-b) + (-a)b = -a(-b + b) = -a0 = 0.

Types of Rings

Definition: Note that a ring does not necessarily have the property of commutative multiplication. When it does, the ring is called a commutative ring.

Definition: 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 unity and is denoted 1. Rings that contain a multiplicative identity are called rings with unity.

Examples

1. $\mathbb{Z}$ is a commutative ring with 1.
2. $M_2(\mathbb{R})$ is a non-commutative ring with $1 = I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$.
3. $2\mathbb{Z}$ is a commutative ring without 1.
4. As a more complicated example, L¹ with convolution is also a commutative ring without 1.

Exercise: Prove that in any ring with 1, the multiplicative identity is unique.

Exercise: As noted earlier, there is a single trivial or zero ring. Show that a ring with unity R is the trivial ring if and only if 1 = 0.

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

Exercise: Prove that $R^\star$ is a group under multiplication.

Definition: An non-zero element $r \in R$ is a zero-divisor when there exists a nonzero $s \in R$ such that rs = 0.

Exercise: Show that a zero-divisor is not a unit.

Definition: A commutative ring R with a $1 \neq 0$ and no zero-divisors is called an integral domain.

Integral domains are modeled after $\mathbb{Z}$, which is where they get their name. They are often very nice rings to work in due to the following theorem:

Theorem (Cancellation Law for Integral Domains): Let R be an integral domain, and let $a,b,c\in R$ be nonzero. Then ab = ac if and only if b = c.

Proof: Evidently ab = ac if b = c. To see the other direction, we rearrange the equality as ab − ac = 0. But then a(b − c) = 0. Since a is nonzero, and R contains no zero divisors, it must be the case that b − c = 0, which is to say that b = c.

Definition: A ring R with a $1 \neq 0$ is a division ring or skew field if all non-zero elements are units i. e. under multiplication, it forms a group with its nonzero elements.

Definition: A field is a commutative division ring. Alternatively, a field F is a ring where $(F,\cdot)$ is an abelian group. As another alternative, a field is an integral domain where all non-zero elements are invertible.

As stated before, integral domains are easy to work with because they are so close to being fields. In fact, the next theorem shows just how close the two are:

Theorem: Let R be a finite integral domain. Then R is a field.

Proof: Let $a\in R$ be nonzero and let $S = \left\{ab| b\in R\right\}$. Clearly S is a subset of R. From the cancellation law, we can see that | S | = | R | (since if two elements ab and ac are equal, then b = c). But then S = R. So then there must be some b such that ab = 1. So a is a unit.

Of course proving that a set with two operations satisfy all of the ring axioms 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 also a ring (under the same two operations as for R). We denote "S is a subring of R" by $S\leq R$.

Theorem: Let S be a subset of a ring R. Then $S\leq R$ iff:

1. S is nonempty
2. S is closed under +
3. $\forall a\in S$, $-a\in S$
4. S is closed under $\cdot$

Examples:

1. $2\mathbb{Z} \leq \mathbb{Z} \leq \mathbb{Q} \leq \mathbb{R} \leq \mathbb{C}$.
2. The trivial ring $\left\{0\right\}$ is a subring of every ring.
3. The set of Gaussian integers $\mathbb{Z}\left[i\right]$ is a subring of the complex numbers $\mathbb{C}$.
4. Let R be a ring, and consider the set of polynomials, R[x]. Then the set of constant polynomials $S = \left\{a_0 | a_0\in R \right\}$ is a subring of R[x]. Note that the elements of R and the constant polynomials in R[x] are not the same objects. However, we can see how we might regard R as a subring of R[x] anyway. We will make this idea more precise when we study homomorphisms in the next section.