20170120

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Problem 14 (What Properties Do Subrings Inherit)

Suppose $R$ is a ring, and $S$ is a subring of $R$.

1. If $R$ is an integral domain, what properties of being an integral domain does $S$ inherit?
2. If $R$ is a field, what properties of being a field does $S$ inherit?
3. Challenge: Find an example of a ring $R$ and a subring $S$ so that both $R$ and $S$ have a unity, but the unity of $S$ is not the same as the unity of $R$.

Let's show that any time we start with an integral domain, then the ring of polynomials with coefficients in that integral domain will still be an integral domain. In other words, let's show that any time you multiply two polynomials and obtain the zero polynomial, then one of the polynomials must have been zero to start with.

Problem 16 (A Polynomial Ring Over An Integral Domain Is An Integral Domain)

If $D$ is an integral domain, show that $D[x]$, the set of all polynomials with coefficients in $D$, is an integral domain.

Problem 17 (Properties Needed For A Factor Group To Be A Factor Ring)

Suppose $R$ is a ring and $A$ is a subring of $R$. Because $A$ is a subgroup of $R$, we know that $R/A$ is an Abelian group, as factor groups of Abelian groups are Abelian. Recall that $$R/A = \{r+A \mid r\in R \},$$ the collection of cosets of $A$.

1. What would it take for $R/A$ to be a ring? What properties must we satisfy? Write out the needed properties.
2. Suppose that $R/A$ is a ring. What additional facts about the cosets must hold to guarantee that $R/A$ is an integral domain?
3. If $R/A$ is an integral domain, what would we need to guarantee that $R/A$ is a field?

Just as we defined a group homomorphism to be a map from one group another that preserved the group operation, we'll define a ring homomorphism to preserve both ring structures.

Definition (Ring Homomorphism And Isomorphism)

A ring homomorphism $\phi$ from a ring $R$ to a ring $S$ is a mapping from $R$ to $S$ that preserves the two ring operations. In other words $\phi(a+b)=\phi(a)+\phi(b)$ and $\phi(ab)=\phi(a)\phi(b)$. A bijective ring homomorphism is called a ring isomorphism.

We've been working with a ring homomorphism since we started studying group theory. This ring homomorphism is precisely the map from $\mathbb{Z}$ to $\mathbb{Z}_n$ obtained through modular arithmetic. The next exercise has you show this. In essence, a ring homomorphism is just a generalization of this map. The kernel of a group homomorphism is the collection of group elements that map to zero. The same holds true for the kernel of a ring homomorphism.

Definition (Kernel Of A Ring Homomorphism)

Let $\phi:R\to S$ be a ring homomorphism. The kernel of $\phi$ is the set $$\ker \phi = \{r\in R\mid \phi(r)=0\}.$$

When studying groups, we invented the word normal subgroup to parallel the properties of a kernel. A subgroup of a group is normal precisely when it is the kernel of a group homomorphism. We then used these properties to create factor groups. We now do the exact same thing with rings. We'll first look at some properties of the kernel, and then we'll turn our attention to factor rings.

Problem 18 (Kernels Are Closed Under Multiplication By Arbitrary Elements)

Let $\phi:R\to S$ be ring homomorphism with kernel $K$. Show the following:

1. The kernel $K$ is a subring of $R$.
2. If $r\in R$ and $k\in K$, then we have $rk\in K$ and $kr\in K$.

The two properties above are precisely the key for characterizing when we can create a factor ring. We use the word ideal to describe these subrings. The next problem shows that you can create a factor ring, provided you have an ideal.

Definition (Ideal)

A subring $A$ of $R$ is called an ideal if $ra\in A$ and $ar\in A$ whenever $a\in A$ and $r\in R$.

Problem 19 (Ideals Give Us Factor Rings)

Let $R$ be a ring and let $A$ be a subring of $R$. Show the following are equivalent.

• The set of cosets $\{ r+A\mid r\in R\}$ is a ring under the operations $(s+A)+(t+A) = (s+t)+A$ and $(s+A)(t+A) = st+A$.
• The subring $A$ is an ideal of $R$.

Definition (Factor Ring)

Let $R$ be a ring and let $A$ be an ideal of $R$. The set of cosets $\{ r+A\mid r\in R\}$ together with the binary operations $(s+A)+(t+A) = (s+t)+A$ and $(s+A)(t+A) = st+A$ is called the factor ring of $R$ by $A$, or just the factor ring $R/A$.

We'll return to factor rings more next time. They happen to be a key tool in our future study, and we'll spend plenty of time becoming comfortable with them. For the rest of today, the next few problems have you practice with some of the definitions we've been building up over the last week, as well as adding in the characteristic of a field and the ideal generated by something.

Problem 20 (Finite Integral Domains Are Fields)

Suppose that $R$ is a finite integral domain. Prove that $R$ is a field.

Problem 21 (Adjoining The Square Root Of A Positive Integer To $\mathbb{Z}_p$)

Show that $\mathbb{Z}_7[\sqrt{3}] = \{a+b\sqrt{3}\mid a,b\in \mathbb{Z}_7\} $ is a field. Given a prime $p$ and positive integer $k$, determine conditions that are both necessary and sufficient for the ring $\mathbb{Z}_p[\sqrt{k}] = \{a+b\sqrt{k}\mid a,b\in \mathbb{Z}_p\}$ to be a field.

Definition (Characteristic Of A Ring)

The characteristic of a ring $R$, written $\text{char } R$, is the smallest positive integer $n$ such that $n\cdot a = a+a+\cdots+a=0$ for all $a\in R$. If no such integer exists, then we say $R$ has characteristic zero.

Problem 22 (Characteristics And Rings With Unity)

Suppose that $R$ is a ring with unity.

1. Show that the characteristic of $R$ is equal to the least positive integer $n$ such that $n\cdot 1=0$, provided the characteristic is not zero.
2. Show that the characteristic of an integral domain is either zero or prime.

Definition (Principle Ideal Generated by $a$, or $\left<a\right>$)

Let $R$ be a commutative ring with unity and let $a\in R$. The set $\left<a\right> = \{ra| \ r\in R\}$ is an ideal of $R$ called the principle ideal generated by $a$.

Definition (Ideal Generated By $a_1, \ldots, a_n$, or $\left<a_1,a_2,\ldots,a_n\right>$)

Let $R$ be a commutative ring with unity, and let $a_1, \ldots, a_n \in R$. The set $I=\left<a_1,a_2,\ldots,a_n\right> = \{r_1a_1+\cdots +r_na_n|r_i\in R \}$ is called the ideal generated by $a_1, \ldots, a_n$. Any other ideal that contains $a_1, \ldots, a_n$ must contain $I$.

Problem 23 (The Ideal Generated By A Subset Is An Ideal)

Let $R$ be a ring and $S=\{a_1, a_2, \ldots a_n\}$ be subset of $R$ consisting of $n$ elements (where $n$ is some positive integer. Prove that the ideal generated by $S$ is actually an ideal of $R$.

For more problems, see AllProblems