Math 361, Spring 2022, Assignment 2

Read:

1. Section 18.

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

1. Ring.
2. Unital (ring).
3. Commutative (ring).

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

1. Rules of sign for rings (this appears in the text as Theorem 18.8).

Solve the following problems:

1. (Direct products of groups) Let $G$ and $H$ be groups. Define a binary operation on the Cartesian product set $G\times H$ by the formula $(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1h_2)$. Show that (i) this operation is associative; (ii) the ordered pair $(e_G,e_H)$ is an identity element for this operation; and (iii) any ordered pair $(g,h)$ has inverse $(g^{-1},h^{-1})$. Thus, with this operation, the Cartesian product $G\times H$ becomes a group, which we call the direct product of the groups $G$ and $H$.
2. List the elements of $\mathbb{Z}_2\times\mathbb{Z}_2$, and then make an operation table for the operation defined above. (Note: with this we have at last proved that the Klein four-group is really a group, in particular that its operation is really associative.)
3. Is the direct product group $\mathbb{Z}_2\times\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_4$? Prove your answer.
4. (Direct products of rings) Now let $R$ and $S$ be rings. Define two binary operations on the Cartesian product set $R\times S$ by the formulas $(r_1,s_1)+(r_2,s_2)=(r_1+r_2,s_1+s_2)$ and $(r_1,s_1)(r_2,s_2)=(r_1r_2,s_1s_2)$. Show that with these operations, $R\times S$ also becomes a ring, also called the direct product of $R$ and $S$.
5. Show that if $R$ and $S$ are both commutative, then so is $R\times S$.
6. Show that if $R$ and $S$ are both unital, then so is $R\times S$.
7. Make a multiplication table for the ring $\mathbb{Z}_2\times\mathbb{Z}_2$, and explicitly identify the unity element of this ring.
8. (Zero-product property fails in direct products) The real number system has the well-known zero-product property: if $xy=0$ then either $x=0$ or $y=0$. Prove that this is not true in arbitrary rings, by giving an explicit counterexample in the ring $\mathbb{Z}_2\times\mathbb{Z}_2$.
9. (Zero-product property also fails for functions) Now consider the ring $\mathrm{Fun}(\mathbb{R},\mathbb{R})$ with the usual "pointwise" operations that we discussed in class. Show that the zero-product property also fails in this ring, by giving two specific non-zero functions $f$ and $g$ with $fg=0$. (Hint: you will almost certainly want $f$ and $g$ to be "piecewise-defined" functions.)
10. (The zero ring) Suppose $R$ is any set with a single element. Show that there is one and only one way of defining binary operations $+$ and $\cdot$ on $R$ which turn $R$ into a ring. Make operation tables for $+$ and $\cdot$. (Any ring with only a single element is called a zero ring. Once we have defined what we mean by an "isomorphism" of rings, we will prove that all zero rings are isomorphic with one another, and because of this we will sometimes speak of the zero ring rather than a zero ring.)
11. (The zero ring is unital) Show that every zero ring is in fact unital. Prove that in any zero ring, one has the surprising equality $0_R=1_R$.
12. (The zero ring is the only ring in which $0_R=1_R$) Suppose now that $R$ is any unital ring in which $0_R=1_R$. Prove that $R$ has only one element, and is thus a zero ring. (Hint: you will need to use at least one of the three assertions in the Rules of Sign theorem referenced above.)

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