Math 361, Spring 2015, Assignment 4

Carefully define the following terms, then give one example and one non-example of each:

1. Kernel (of a ring homomorphism).
2. Ideal.
3. Principal idea.
4. Quotient ring (a.k.a. factor ring).
5. Prime ideal.
6. Maximal ideal.

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning images and pre-images of subrings.
2. Theorem characterizing the subrings that can be kernels of homomorphisms.
3. Fundamental theorem of ring homomorphisms.
4. Equivalent conditions for containment of principal ideals.
5. Equivalent condition for equality of principal ideals.
6. Theorem characterizing prime ideals (in terms of properties of the corresponding quotient rings).
7. Theorem characterizing maximal ideals (in terms of properties of the corresponding quotient rings).
8. Theorem relating maximal ideals to prime ideals.

Solve the following problems:

1. Section 26, problems 3, 11, 12, 13, and 14.
2. Section 27, problems 1, 10, 15, 16, and 17.

--------------------End of assignment--------------------

Questions:

Does anyone know the theorem number corresponding to "Theorem characterizing the subrings that can be kernels of homomorphisms" in the book?

The book never quite comes out and says it, but what I had in mind is this: kernels must be ideals, and every ideal is the kernel of some homomorphism. -Steven.Jackson (talk) 22:02, 23 February 2015 (EST)

Solutions:

Definitions:

1. The Kernel of a ring morphism $\phi$ is the subring $S$ such that $\phi[S]=\left\{0\right\}$.
2. An Ideal $I$ of a ring $R$ is a subring of $R$ such that for all $i\in I, a \in R$, $ai\in R$ and $ia \in R$, that is to say $I$ preserves right and left products.
3. If $R$ is a commutative unital ring and $a \in R$, the ideal $\{ra | r \in R \}$ of all multiples of $a$ is the Principal Ideal Generated By $a$ and is denoted by $\langle a \rangle$. An ideal $I$ of $R$ is a Principal Ideal if $I=\langle a \rangle$ for some $a \in R$.
4. Let $I$ be an ideal of a ring $R$. Then the additive cosets of $I$ form the Quotient Ring $R/I$ under the binary operations defined by:
   1. $(a+I)+(b+I)=(a+b)+I$
   2. $(a+I)(b+I)=ab+I$
5. An ideal $I\ne R$ in a commutative ring $R$ is a Prime Ideal if $ab\in I$ implies $a\in I$ or $b\in I$ for $a,b\in R$.
6. A Maximal Ideal of a ring $R$ is an ideal $M\ne R$ such that there is no proper ideal $I$ of $R$ with $I$ properly conaining $M$.

Theorems:

1. Let $S,R$ be rings and let $\phi: R \rightarrow S$ be a ring morphism. Let $A$ be a subring of $R$ and let $B$ be a subring of $S$. Then:
   1. $\phi[A]$ is a subring of $S$ and
   2. $\phi^{-1}[B]$ is a subring of $R$
3. Let $\phi:R \rightarrow R'$ be a ring morphism with kernel $N$. Then $\phi[R]$ is a ring, and the map $\mu:R/N \rightarrow \phi[R]$ given by $\mu(x+N)=\phi(x)$ is an isomorphism. If $\gamma: R \rightarrow R/N$ is the morphism given by $\gamma(x)=x+N$, then for each $x\in R$ we have $\phi(x)=\mu\gamma(x)$.
4. Let $D$ be a domain. The following statements are equivalent for all $a,b\in D$:
   1. $\langle a \rangle \subseteq \langle b \rangle$
   2. $a \in \langle b \rangle$
   3. $b|a$
5. $\langle a \rangle = \langle b \rangle$ if and only if $a\sim b$.
6. An ideal $I$ of a ring $R$ is prime if and only if $R/I$ is a domain.
7. An ideal $I$ of a ring $R$ is maximal if and only if $R/I$ is a field.
8. Evey maximal ideal is a prime ideal.

Problems:

26.3:

$\mathbb{Z}_{12}/\langle 1 \rangle$ is isomorphic to $\{0\}$

$\mathbb{Z}_{12}/\langle 2 \rangle$ is isomorphic to $\mathbb{Z}_{6}$

$\mathbb{Z}_{12}/\langle 3 \rangle$ is isomorphic to $\mathbb{Z}_{4}$

$\mathbb{Z}_{12}/\langle 4 \rangle$ is isomorphic to $\mathbb{Z}_{3}$

$\mathbb{Z}_{12}/\langle 6 \rangle$ is isomorphic to $\mathbb{Z}_{2}$

$\mathbb{Z}_{12}/\{0\}$ is isomorphic to $\mathbb{Z}_{12}$

26.11:

They are not of interest. $R/R$ is trivial and $R/\{0\}$ is isomorphic to $R$.

26.12:

$\mathbb{R}$ is an integral domain and $\mathbb{R}/\{0\}$ is a field.

26.13:

$\mathbb{Z}$ is an integral domain and $\mathbb{Z}/\langle 10 \rangle$ has zero divisors (i.e. $5+\langle 10 \rangle$).

26.14:

$\mathbb{Z}_{6}$ has 0 divisors and $\mathbb{Z}_6/\langle 2 \rangle$ is an integral domain.

27.1:

Prime ideals: $\langle 2 \rangle, \langle 3 \rangle$. Maximal ideals: $\langle 2 \rangle, \langle 3 \rangle$.

27.10:

A maximal ideal of a ring $R$ is an ideal $M$ such that there is no ideal $N$ of $R$ such that $M\subsetneq N \subsetneq R$.

27.15:

$\mathbb{Z}\times 3\mathbb{Z}$ is a maximal ideal of $\mathbb{Z} \times \mathbb{Z}$.

27.16:

$\mathbb{Z}\times \{0\}$ is a prime ideal of $\mathbb{Z} \times \mathbb{Z}$ that is not maximal.

27.17:

$\mathbb{Z}\times 10\mathbb{Z}$ is a non-trivial proper ideal of $\mathbb{Z} \times \mathbb{Z}$ that is not prime.