Ring Theory 3 Intuition behind Subrings HD

Defining and investigating what it means to be a subring of a ring. This video examines the intuition behind subrings, and the next video will cover an example proof. Practice problems: 1) Which of the following are subrings of R? If it is a subring, prove it. If not, state which step of the subring test the set fails. a) {a + b*sqrt(5), where a,b are integers} b) {a + b*cuberoot(5), where a,b are integers} c) 3Z union 5Z d) 3Z intersect 5Z 2) Is Z_5 a subring of Z_10? 3) Let R be a ring, and let S and T be subrings of R. Prove that S intersect T is a subring of R. 4) Let R be a ring, and let x be a fixed element of R. Prove that A = {all r in R | rx=0} is a subring of R. 5) (Challenge) Let R be a commutative ring. Prove that N = {all x in R | x^k=0 for some natural number k} is a subring of R. Note: k may not be the same integer for every element of N. Partial solutions will be posted in the description of the next video.