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### Homework 2

- Section 14.7
- Homework: 1, 2, 10
- Hand In: 2

- Section 14.8
- Homework: 3, 4, 6, 7
- Hand In: 4, 6

- Section 10.1
- Homework: 3, 4, 8, 13, 21 and the following problems:
- 24. Suppose that \(R\) is a commutative ring with unity. A (left) module is
*irreducible*or*simple*if it is not the zero module and if it has no proper submodule.- Prove that any irreducible module is isomorphic to \(R/M\), where \(M\) is a maximal ideal of \(R\).
- Let \(\varphi : S \to S'\) be a homomorphism of irreducible \(R\)-modules. Prove that either \(\varphi\) is the zero map or that \(\varphi\) is an isomorphism.

- 25. The
*annihilator*of a (left) \(R\)-module \(V\) is the set \(I = \{r \in R \vert rV = 0\}\).- Prove that \(I\) is a (2-sided) ideal of \(R\)
- Find the annihilator of the \({\mathbb Z}\)-module \({\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/3{\mathbb Z}\times {\mathbb Z}/4{\mathbb Z}\).

- 24. Suppose that \(R\) is a commutative ring with unity. A (left) module is
- Hand In: 24, 25

- Homework: 3, 4, 8, 13, 21 and the following problems:
- Section 10.2
- Homework: 3, 4, 5, 6, 9, 10 and the following problem:
- 15. Prove that the ring of endomorphisms of an irreducible module over a commutative ring with unity is a division ring. (Source: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991))

- Hand In: 15

- Homework: 3, 4, 5, 6, 9, 10 and the following problem:
- Section 10.3
- Homework: 1, 4, 9, 10, 11 and the following problem:
- 28. Let \(R\) be a commutative ring with unity. Let \(I\) be a nonzero ideal of \(R\). Prove that \(I\) is a free \(R\)-module if and only if it is a principal ideal of \(R\) and is generated by an element that is not a zero divisor in \(R\). (Source: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991))

- Hand In: 28

- Homework: 1, 4, 9, 10, 11 and the following problem:
- Section 12.1
- Homework: 1, 2, 6, 8 and the following problem:
- 23. Let \(R\) be an integral domain with field of quotients \(F\) (so \(R\ \subset F\)). Regard \(F\) as an \(R\)-module via the field operations. Prove that \(F\) is torsion-free as an \(R\)-module, while \(F/R\) is a torsion module. (Source: Derek J.S. Robinson, "Abstract Algebra (Second Edition)", De Gruyter, Berlin/Boston (2015))

- Hand In: 23

- Homework: 1, 2, 6, 8 and the following problem: