Ring Theory 4: Subring Proof Example

Example of proving a subset of a ring is, itself, a ring. Partial solutions to problems of video 3: 1) a) This is a subring of R. Simply prove the four properties. b) This is not a subring of R, since it is not closed under multiplication. For example, 5^(1/3) is in the set. So 5^(1/3) * 5^(1/3) = 5^(2/3) should also be in the set, but there is no way to make this element from a + b*5^(1/3) if a,b are integers. c) This is not a subring of R, since it is not closed under addition. For example, 3 is in 3Z, and 5 is in 5Z. Therefore both 3 and 5 are in 3Z union 5Z. However, 3 + 5 = 8 is not in the union, since 8 is not a multiple of 3 or 5. d) This is a subring of R. Simply prove the four properties. 2) Z_5 is NOT a subring of Z_10. Since the operations of Z_5 are done modulo 5, it does not share the same operations of Z_10 (modulo 10), and therefore cannot be a subring. The remaining problems are proofs, which I do not have room to post here. If you would like a specific problem proved, simply request it in the comments below and I may make a video detailing the proof.