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• Do the following problems:

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.

1. 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.

1. 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.)

1. 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))

1. 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].

1. 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.