MATH0021 Sample Exam
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH0021
Answer all questions .
Throughout R will denote a commutative ring
1. Let a e R _ {0}; explain what is meant by saying that a is irreducible.
i) Suppose x e 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 + e ^_15) is irreducible in 勿[^_15] if e = 土1.
iii) By evaluating (1 + ^_15)(1 _ ^_15) or otherwise show that 勿[^_15] is not a principal ideal domain.
(20 marks)
2. Let G = (0 → A 
B 
C → 0) be a short exact sequence of R modules, Explain what is meant by saying that
a) G splits on the right ; b) G splits.
Prove that if G splits then G splits on the right.
i) State Schanuel’s Lemma.
ii) Let
, 0 → I → P2 → P1 → P0 → M → 0;
．
( 0 → J → Q2 → Q1 → Q0 → M → 0
be exact sequences of R-modules where each Pr , Qs is finitely generated and pro- jective. Show there exist finitely generated projective R-modules P and Q such that
I 2 P 
J 2 Q
and express P , Q in terms of Pr and Qs .
iv) Hence show that Hk (I; N) 
Hk (J; N) for k > 0 and all R-modules N .
(25 marks)
3. If M is a module over the integral domain R explain what is meant by the torsion submodule Θ(M) of M .
If a e R _ {0} the set Θa (M) is defined by
Θa (M) = {m e M I m . a = 0}.
i) Show that Θa (M) is an R-submodule of Θ(M).
ii) If Θ(M) is finitely generated and R is an integral domain show that there exists a e R _ {0} such that Θa (M) = Θ(M).
iii) Give an example to show that the conclusion of ii) is false if M is not finitely generated.
If M is finitely generated and the module M/Θ(M) is projective then, without making any further assumption on R, show that Θ(M) is finitely generated.
(25 marks)
4. Define the global dimension gd(R) of the ring R.
i) If R is a principal ideal domain which is not a field show that gd(R) = 1.
ii) Let 勿[t] be the ring of polynomials in a single variable t over 勿, let p e 勿 be prime and let Fp be the field with p elements. Show that the following sequence of 勿[t]-modules is exact;
0 → 勿[t] (
) 勿[t] 2 勿[t] (p,t_
1) 勿[t] 
Fp → 0
where η : 勿[t] → Fp is the unique ring homomorphism determined by η(t) = 1.
iii) Use this sequence to compute Hk (Fp , 勿[t]) for all k ~ 0.
iv) Hence show that 勿[t] is not a principal ideal domain.
(30 marks)
2023-04-21