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Do the following problems: Let R be a ring. Prove that the following are equivalent statements about R: For all a, b ∈ R, if ab = 0 then a = 0 or b = 0. For all a, b, c ∈ R, if ab = ac and a 6= 0 then…
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.
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.)
A1(X) + B1(X) ≡ A2(X) + B2(X)
(mod P (X))
and
A1(X) · B1(X) ≡ A2(X) · B2(X)
(mod P (X))
such that
A(X) · P (X) + B(X) · Q(X) = 1.
Do Problems 1, 3, 5, 6, 7 and 8 at the end of chapter 7 (section 7.6). Postponed until next week.