Q.1) Classify the following statements as a true or false statements: Q a. A simple ring R is simple as a right R-module. b. Every ideal of ZZ is small ideal. very den to is lovaginz c. A nontrivial direct summand of a module cannot be large or small submodule. d. The sum of a finite family of small submodules of a module M is small in M. e. The direct product of a finite family of projective modules is projective f. The sum of a finite family of large submodules of a module M is large in M. g. Zz contains no minimal submodules. h. Qz has no minimal and no maximal submodules. i. Every divisible Z-module is injective. j. Every projective module is a free module. a homomorp cements Q.4) Give an example and explain your claim in each case: a) A module M which has a largest proper submodule, is directly indecomposable. b) A free subset of a module. c) A finite free module. d) A module contains no a direct summand. e) A short split exact sequence of modules.

Exercises If and are two ideals of the ring , prove that is an ideal of .

Show that the ideal is a maximal ideal of .

27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.

. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .

29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .

Label each of the following statements as either true or false. The only ideal of a ring R that property contains a maximal ideal is the ideal R.

12. Let be a commutative ring with unity. If prove that is an ideal of.

Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.

In the ring of integers, prove that every subring is an ideal.