MATH0021
Answer all questions.
Throughout R will denote a commutative ring
1. Let a ∈ R {0}; explain what is meant by saying that a is irreducible.
i) Suppose x ∈ R {0} satisfies
x = a1 . . . . . . am = b1 . . . . . . bn
where each ai
, bj is irreducible. If R is a principal ideal domain show that m = n.
ii) Show that (1 + _x000f_ √
15) is irreducible in Z[
√
15] if = ±1.
iii) By evaluating (1 + √
15)(1
√
15) or otherwise show that Z[
√
15] is
not a principal ideal domain.
(20 marks)
2. Let S = (0 → A
i
→ B
p→ C → 0) be a short exact sequence of R modules,
Explain what is meant by saying that
a) S splits on the right ; b) S splits.
Prove that if S splits then S splits on the right.
i) State Schanuel’s Lemma.
ii) Let
