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Similarly, the sum of two left [resp. right] ideals is a left [resp. right] ideal. 2 1. What are the ideals in the ring of integers? 2. Let Mn (R) be the ring of n by n matrices with coeﬃcients in the ring R. If Ck is the subset of Mn (R) consisting of matrices that are 0 except perhaps in column k, show that Ck is a left ideal of Mn (R). Similarly, if Rk consists of matrices that are 0 except perhaps in row k, then Rk is a right ideal of Mn (R). 3. In Problem 2, assume that R is a division ring, and let Eij be the matrix with 1 in row i, column j, and 0’s elsewhere.

Since R is an integral domain, the only possible roots of f are a1 , . . , ak . 4 Example Let R = Z8 , which is not an integral domain. The polynomial f (X) = X 3 has four roots in R, namely 0, 2, 4 and 6. 5 In Problems 1-4, we review the Euclidean algorithm. Let a and b be positive integers, with a > b. Divide a by b to obtain a = bq1 + r1 with 0 ≤ r1 < b, then divide b by r1 to get b = r1 q2 + r2 with 0 ≤ r2 < r1 , and continue in this fashion until the process terminates: r 1 = r 2 q 3 + r 3 , 0 ≤ r 3 < r2 , ..

Therefore the prime factors of ai+1 consist of some (or all) of the prime factors of ai . Multiplicity is taken into account here; for example, if p3 is a factor of ai , then pk will be a factor of ai+1 for some k ∈ {0, 1, 2, 3}. Since a1 has only ﬁnitely many prime factors, there will come a time when the prime factors are the same from that point on, that is, an = an+1 = . . for some n. (2) Let a1 be any nonzero element. If a1 is irreducible, we are ﬁnished, so let a1 = a2 b2 where neither a2 nor b2 is a unit.