Math 361, Spring 2020, Assignment 2

Read:

1. Section 26.

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

1. Subring.
2. Unital subring (be careful!).
3. Ring homomorphism.
4. Unital ring homomorphism.
5. Image (of a ring homomorphism).
6. Kernel (of a ring homomorphism).
7. Ideal.
8. $R/I$ (the quotient of a ring $R$ by an ideal $I$; the textbook calls it the factor ring of $R$ by $I$).
9. Canonical projection (from $R$ to $R/I$).
10. $\left\langle a\right\rangle$ (the principal ideal generated by an element $a$ in a commutative ring $R$).

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

1. Laws of sign.
2. Theorem concerning pushforwards and pullbacks of subrings.
3. Theorem concerning absorbtion of products by kernels.
4. Theorem concerning the definition of coset multiplication.

Solve the following problems:

1. Section 26, problems 1, 3, 4, 8, 9, 11, 15, 17, and 27.
2. Suppose that $R$ is a unital ring and that $\phi:R\rightarrow S$ is an epimorphism. Show that $S$ is unital, and that $\phi(1_R)=1_S$. Why doesn't problem 9 contradict this result?

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