Mathematics 351 – Abstract Algebra I Discussion 1 – More Examples of Rings September 3, 2007

Background

Last time, we introduced the general definition of a ring, an nonempty set R with two binary operations +, · satisfying the following 8 axioms:

1. (closure under +) For all a, b ∈ R, a + b ∈ R.

2. (associativity of +) For all a, b, c ∈ R, (a + b) + c = a + (b + c).

3. (commutativity of +) For all a, b ∈ R, a + b = b + a.

4. (additive identity) There is an element 0 ∈ R such that a + 0 = a for all a ∈ R.

5. (additive inverses) For each a ∈ R, there is an element x = −a ∈ R such that a+x = 0.

6. (closure under ·) For all a, b ∈ R, a · b ∈ R.

7. (associativity of ·) For all a, b, c ∈ R, (a · b) · c = a · (b · c).

8. (distributive laws) For all a, b, c ∈ R, a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c.

If, in addition there is a multiplicative identity 1 in R, we call R a ring with identity. If the multiplication operation · is also commutative, we say R is a commutative ring.

In a ring R, a zero divisor is an element a ≠ 0 ∈ R such that there exists some b ≠ 0 with a · b = 0 or b · a = 0. An integral domain is a commutative ring with identity that has no zero divisors (i.e. where a ≠ 0 and b ≠ 0 implies a · b ≠ 0).

Finally, a commutative ring with identity 1 ≠ 0, in which every nonzero element has a multiplicative inverse (that is, for each a ≠ 0, the equation a·x = 1 has a solution x = a^{−1} in the ring) is called a field.

Discussion Questions

I. Let Q denote the field of rational numbers, and let p be a (fixed) prime number in Z ⊂ Q. Let

R_{p} = \{\frac{a}{b} ∈ Q : a, b ∈ Z and b = p^{i} for some i ≥ 0\} .

A) Show carefully that R_{p} is a commutative ring with identity, using the usual sum and product operations from Q. Note: Many of the ring properties in R_{p} will follow from corresponding statements in Q, because R_{p} is a subset of the ring Q, so if you are thinking about it the right way, this should not be as tedious as it might appear at first(!)

B) Is R_{p} an integral domain? Is R_{p} a field? Prove your assertions. C) The ring R_{p} here is an example of what is called a subring of the ring Q. What should

it mean to say a ring S is a subring of a ring R in general?

II. Let L = {a ∈ R : a > 0}. We can define new “sum” and “product” operations on L as follows:

a ⊕ b = ab  and  a ⊗ b = a^{ln(b)}.

Is L is a ring under these operations? Why or why not? Is it a subring of the ring (R, +, ·) (+, · = the usual sum and product of reals)?

III. In class last time, we saw that

C^{1}(R) = {f : R → R : f′ exists and is continuous at all x ∈ R}

is a commutative ring with identity.

A) Let

f(x) = \{ \begin{matrix}3x^{2} − 2x^{3} & if 0 ≤ x ≤ 1 \\ 2x^{3} − 9x^{2} + 12x − 4 & if 1 ≤ x ≤ 2 \\ 0 & otherwise \end{matrix}

Sketch the graph of f and show that f is an element of C^{1}(R). B) Is C^{1}(R) an integral domain? (Hint: You can make lots of other functions in C^{1}(R)

by “doing things to” the function f from part A.)

IV. Let (R, +_{R}, ·_{R}), (S, +_{S}, ·_{S}) be any two rings. We can define sum and product operations +, · “component-wise” on the Cartesian product:

R × S = {(a, b) : a ∈ R, b ∈ S}

by the following rules:

\begin{aligned}(a, b) + (a′, b′) & = (a +_{R} a′, b +_{S} b′) \\ (a, b) · (a′, b′) & = (a ·_{R} a′, b ·_{S} b′). \end{aligned}

A) Show that the above operations always make R × S into a ring. B) Let R = Z_{2} with addition and multiplication mod 2, and let S = Z_{3} with addition

and multiplication mod 3. Make addition and multiplication tables for the operations +, × on Z_{2} × Z_{3} defined as above.

C) Is Z_{2} × Z_{3} “the same” as some other ring we know? In what sense?

Assignment

Group writeups due on Monday, September 10.