Math 361, Spring 2021, Assignment 2

Read:

1. Section 14.

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

1. Ring.
2. Zero element (of a ring).
3. Opposite (of an element of a ring).
4. Unity element (of a unital ring).
5. Inverse (of an invertible element of a ring with unity).

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

1. Theorem relating kernels to injectivity.
2. Theorem concerning the normality of kernels.
3. Theorem characterizing which subgroups can be kernels of homomorphisms.
4. Fundamental theorem of group homomorphisms.

Solve the following problems:

1. Section 13, problems 49 and 51.
2. Section 14, problems 24 and 31.
3. Section 18, problems 1, 3, 7, 8, 11, and 13 (in these problems, simply determine whether the given structure is a ring, whether it is commutative, and whether it has unity; we will discuss fields next week).
4. Let $\phi$ and $\widehat{\phi}$ be as in the Fundamental Theorem of Homomorphisms. Prove that $\mathrm{im}\,\phi=\mathrm{im}\,\widehat{\phi}$.

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