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Math 366 Assignment 4 Due Wednesday, December 1 NOTE: I stepped up the theory a bit on this one, though none of the arguments are terribly long. You have two weeks to work on it, and I’ll be happy to talk about it in class on the Monday before it is due. I will cover fields Monday, Wednesday, and Friday, going over some basic problems about fields (in class, not assigned as homework) on Friday (12/3) or Monday (12/6). I will hand out the final exam one of those two days. The exam problems will be similar to the problems in this assignment, with a problem or two thrown in about fields, depending on how much we manage to cover before the exam.... 1. The ring {0, 2, 4, 6, 8} (using mod 10 addition and multiplication) has a unity. Find it, and demonstrate why it is in fact the unity element (i.e., write down the 5 arithmetic statements that confirm this). 2. Prove that the intersection of any finite collection of subrings or a ring R is a subring. In particular, suppose Si is a subring of R for i = 1, . . . , N . Use the fact that these sets are all subrings to argue why a − b ∈ ∩Si for any a, b ∈ ∩Si . 3. Recall that a|b means that a divides b. (a) In Z6 , show that 4|2; (b) in Z8 , show that 3|7; and (c) in Z1 5, show that 9|12. 4. A ring element a is called idempotent if a2 = a. Prove that the only idempotents in an integral domain are 0 and 1. 5. Find both a zero-divisor and an idempotent (two different elements) in Z5 [i] = {a + bi | a, b ∈ Z5 }. (Recall that i2 = −1). 6. If a is an idempotent in Zn , show that 1 − a is also an idempotent. 7. Let S = {a + bi | a, b ∈ Z, b is even}. Show that S is not an ideal of Z[i]. 8. If an ideal I of a ring R contains a unit, show that I = R. 9. Show that the map φ : Z5 → Z10 sending x to 5x does not preserve addition (and is thus not a homomorphism). 10. Let ψ : R[x] → R be defined by sending f (x) to f (1) (where R[x] is the ring of polynomials with real coefficients). Describe the kernel of ψ. NOTE: Much of this is from the 6th edition of Gallian’s Contemporary Abstract Algebra.