计算机科学的数学表达与推理代写 CSC 165 H1代写 数学代写
367CSC 165 H1 Term Test 3 — Question 1 of 4 计算机科学的数学表达与推理代写 Aids Allowed: Your own notes taken during lectures and office hours, the lecture slides and recordings (for all secti...
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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)
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.
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.
Prove that for every ring R, the following are equivalent.
(a) Every R-module is projective.
(b) Every R-module is injective.
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.
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CSC 165 H1 Term Test 3 — Question 1 of 4 计算机科学的数学表达与推理代写 Aids Allowed: Your own notes taken during lectures and office hours, the lecture slides and recordings (for all secti...
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View detailsCSC165H1 Problem Set 3 数学表达与推理代写 General instructions Please read the following instructions carefully before starting the problem set. They contain important information about ...
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