## Math 5120 Graduate Algebra II

### Homework 2

Due Friday March 8, in class
• 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.
1. Prove that any irreducible module is isomorphic to $$R/M$$, where $$M$$ is a maximal ideal of $$R$$.
2. 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.
(Source: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991))
• 25. The annihilator of a (left) $$R$$-module $$V$$ is the set $$I = \{r \in R \vert rV = 0\}$$.
1. Prove that $$I$$ is a (2-sided) ideal of $$R$$
2. Find the annihilator of the $${\mathbb Z}$$-module $${\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/3{\mathbb Z}\times {\mathbb Z}/4{\mathbb Z}$$.
(Source: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991))
• Hand In: 24, 25
• 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
• 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
• 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