MAT401 Problem Set 1 Summer 2022
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MAT401
Summer 2022
Problem Set 1
1. Textbook Exercise 9.
2. (a) Let Rl , R2 , . . . , Rn be rings. Construct the direct sum
Rl e R2 e . . . e Rn = {(rl , r2 , . . . , rn ) l ri e Ri } Define componentwise addition and multiplication on this set. That is,
(rl , r2 , . . . , rn ) e (sl , s2 , . . . , sn ) = (rl + sl , r2 + s2 , . . . , rn + sn ) and (rl , r2 , . . . , rn ) o (sl , s2 , . . . , sn ) = (rl . sl , r2 . s2 , . . . , rn . sn )
Show that (Rl e R2 e . . . e Rn , e , o) is a ring.
(b) Prove or give counter example : If Dl and D2 are domains, then so is Dl e D2 .
3. Show that R is a domain if and only if is R[x] is a domain.
4. (a) Recall that we denote degree of a polynomial f(x) by ∂(f). Show that if R is a
domain, then for any two non-zero polynomials f(x), g(x) e R[x], ∂(fg) = ∂(f) + ∂(g)
(b) Give a counter example to show that the above statement need not hold if R is
not a domain.
5. (a) Show that every non-zero element in zn is either a unit or a zero-divisor.
(b) Give an example of a ring and a non-zero element in it, which is neither a
zero-divisor nor a unit.
2022-05-21