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1. Let R be a ring. Prove that the following are equivalent statements about R:

(1) For all a, b ∈ R, if ab = 0 then a = 0 or b = 0.

(2) For all a, b, c ∈ R, if ab = ac and a 6= 0 then b = c.

In other words, prove that if (1) is true for R, then so is (2), and that if (2) is

true for R, then so is (1).

Note that (1) is the same as saying that R has no zero divisors, and is sometimes

called the “zero product property”. (2) is the “multiplicative cancellation law”. So

what you have proved is that R has no zero divisors iff the zero product property

is true in R iff the cancellation law works in R.

2. Let F be a field, and let P(X) ∈ F[X] (that is, P(X) is a polynomial with

coefficients in F). Recall that in class we defined, for A(X), B(X) ∈ F[X],

A(X) ≡ B(X) (mod P(X)) if P(X) | (A(X) − B(X)).

Prove that this is an equivalence relation. (Remember that this means you must

show that it is reflexive, symmetric, and transitive.)

3. Again let F be a field, and let P(X) ∈ F[X]. Assume A1(X) ≡ A2(X) (mod P(X))

and B1(X) ≡ B2(X) (mod P(X)). Show that

A1(X) + B1(X) ≡ A2(X) + B2(X) (mod P(X)) and

A1(X) · B1(X) ≡ A2(X) · B2(X) (mod P(X))

4. Use the Euclidean algorithm for polynomials to find the greatest common divisor

of 4X3 − 4X2 − 3X + 2 and 8X4 − 12X3 + 8X − 3 in R[X].

5. Let A(X) = X3 + 2X + 2 and B(X) = X2 + 3X + 4 in F5[X]. Use the extended

Euclidean algorithm for polynomials to find polynomials P(X), Q(X) ∈ F5[X]

such that

A(X) · P(X) + B(X) · Q(X) = 1.

• Do Problems 21, 22, 33 and 34 at the end of chapter 3 (section 3.13).

• Do Problems 1, 3, 5, 6, 7 and 8 at the end of chapter 7 (section 7.6). Postponed until

next week.

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