MT4512 Automata, Languages and Complexity 数学代码代写 EXAM DURATION: 2 hours EXAM INSTRUCTIONS: Attempt ALL questions. The number in square brackets shows the maximum marks obtainable for tha...View details
MATH 5735 – Modules and Representation Theory
模块和表示论代写 1. (9 marks) Recall that an integral domain is a commutative ring (with unity) that has no zero divisors. (a) Prove that if R is an integral domain
(Due Friday, 18 March, 2022, 8pm)
1. (9 marks) 模块和表示论代写
Recall that an integral domain is a commutative ring (with unity) that has no zero divisors.
(a) Prove that if R is an integral domain, then the set of torsion elements in an R-module M (denoted Tor(M)) is a submodule of M.
(b) Give an example of a ring R and an R-module M such that Tor(M) is not a submodule.
(c) Show that if R has zero divisors, then every non-zero R-module has non-zero torsion elements.
2. (9 marks) 模块和表示论代写
Let R be a commutative ring and M an R-module.
(a) Show that HomR(R, M) can be given the structure of an R-module in a natural way. (Define this R-module structure explicitly, check that the structure you wrote down is well-defined, then check that it satisfies the axioms of an R-module.)
(b) Show that HomR(R, M) and M are isomorphic as R-modules.
(c) Show that EndR(R) and R are isomorphic as rings.
4. (5 marks) 模块和表示论代写
Prove that for every ring R, the following are equivalent.
(a) Every R-module is projective.
(b) Every R-module is injective.
5. (8 marks) 模块和表示论代写
Consider the C[x]-module M := C[x]/(x3 − x2 ), where (x3 − x2 ) is the ideal in C[x] generated by x3 − x2 .
(a) Show that M is finite length by constructing a composition series.
(b) Write down the composition factors of M.
(c) Is M Noetherian? Justify your answer.
(d) Is M Artinian? Justify your answer.