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.

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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 $a \in R$.